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EICHAED A. PROCTOR, B.A. Cimbkidge, "" 

Honorary Secretary of the Royal Astronomical Society ; 




'With how sad steps, O Moon, thou climb'stthe sky, 
How silently and with how wan a face !" Wordsworth. 

" Art thou pale for weariness 
Of climbing heaven and gazing on the earth. 

Wandering companionless 
Among the stars that have a different birth, 
And ever changing, like a joyless eye 
That finds no object worth its constancy ?" Shellby. 


(enlarged by brothers) 



{All rights reserved.) 

\ iP *^ ^W'Sli^ Ato^M^S, PEINTEE3, 

LONDON, i't^.C. 

'^SIJY Of -fO!? 



D.C.L., F.R.S., F.R.A.S., &c. 








Although I had long purposed to draw up a treatise 
on tlie MooUj to form part of the series of volumes to 
which my works on Saturn and the Sun appertain. I 
originally proposed that this treatise should be the 
last instead of the third of that series. But Mr. 
Brothers being desirous of publishing three of Mr. 
Rutherfurd^s magnificent lunar photographs,, asked me 
to prepare some accompanying letterpress^ and the 
present work thus had its origin ; for I soon found 
that if I supplied the required quantity of letterpress, 
I should to some extent injure the prospects of the 
more complete work which I had in view. Moreover, 
it seemed to me desirable not to take up the subject 
partially and resume it at some distant epoch, but to 
deal with it in a single effort. This will serve to 
explain the delay which ensued ; for certain parts of 
my subject-matter required much time and close 
application, for their thorough and independent in- 
vestigation. I ought also (for reasons which will 
be understood by subscribers to the large volume) to 
explain that Mr. Brothers and I found it convenient 
to separate our interests; so that while I provide 


the octavo volume (witliout photographs) to accom- 
pany his large volume of photographs (see p. 400), he 
provides the smaller photographs for the complete 
octavo treatise, on pre-arranged terms. 

I take this opportunity of thanking Mr. Eutherfurd, 
on Mr. Brothers's behalf and my own, for his liberality 
in permitting the publication of his admirable photo- 
graphs. I feel that the prospects of this work's 
success have been greatly enhanced, also, by the 
skill with which Mr. Brothers has enlarged these 
beautiful pictures of our satellite. (See advertisement 
leaf, p. 400.) 

While this treatise has been in progress,! have heard 
that a work on the Moon has been commenced under 
the supervision of a well-known student of the Moon ; 
and my friend Mr. Webb has mentioned his own 
intention of writing one day a book upon the same 
subject. It seemed to me, therefore, desirable, while 
presenting so much as was necessary for the complete- 
ness of the present work on the parts of the subject 
which the promised works are severally likely to treat 
with special fulness, to devote my chief attention to 
those departments to which my own tastes chiefly 
invited me. It is perhaps hardly necessary to remark 
that in any case some portions of the subject must 
have been less fully treated of than others, and some 
omitted altogether, since, in fact, ten such volumes 
as the present Would be insufficient to deal satisfac- 
torily with all the matters of interest connected with 
the Moon. 


The only chapters in this treatise which require 
comment here, are the second and third, relating to 
the motions of the Moon, and to her changes of 
aspect : 

In Chapter II. I have given a very full account of 
the peculiarities of the Moon's motions ; and notwith- 
standing the acknowledged difficulty of the subject, I 
think my account is sufficiently clear and simple to be 
understood by any one (even though not acquainted 
with the elements of mathematics) who will be at the 
pains to read it attentively through. I have sought 
to make the subject clear to a far wider range of readers 
than the class for which Sir G. Airy's treatise on 
gravitation was written, while yet not omitting any 
essential points in the argument. In order to combine 
independence of treatment with exactness and com- 
pleteness, I first wrote the chapter without consulting 
any other work. Then I went through it afresh, care- 
fully comparing each section with the corresponding 
part of Sir G. A.iry's Gravitation, and Sir J. HerschePs 
chapters on the lunar motions in his " Outlines of 
Astronomy .'' I was thus able to correct any errors in 
my own work, while in turn I detected a few (mentioned 
in the notes) in the works referred to. 

I have adopted a much more complete and exact 
system of illustration in dealing with the Moon's 
motions than either of my predecessors in the expla- 
nation of this subject. I attach great importance to 
this feature of my explanation, experience having 
satisfied me not only that such matters should be very 


freely illustrated, but that the illustrations should aim 
at correctness of detail, and (wherever practicable) 
of scale also. Some features, as the advance of the 
perigee and the retreat of the nodes, have, I believe, 
never before been illustrated at all. 

In Chapter III. I give, amongst other matters, a full 
explanation of the effects due to the lunar librations. 
I have been surprised to find how imperfectly this 
interesting and important subject has been dealt with 
hitherto. In fact, I have sought in vain for any dis- 
cussion of the subject with which to compare my own 
results. I have, however, in various ways sufficiently 
tested these results. 

The table of lunar elements will be found more 
complete than that usually given. In fact, in this 
table, and throughout the work, my aim has been to 
help the student of the subject by supplying informa- 
tion not given, or not so completely given, elsewhere. 
It has always seemed to me that although in works on 
scientific snbjects much of what is written must be 
common property and many facts must be compiled 
from the writings of other authors, the main purpose 
of the writer should be to present results which he has 
himself worked out and which are calculated to be of 
use to others. I doubt, indeed, whether any one is 
justified in writing a treatise on science unless he has 
such a purpose chiefly in view. 

London: Jwne 1873. 



The Moon's Distance, Size, and Mass. 


Introduction 1 

Order in which lunar phenomena were probably discovered 2 

Determination of the moon's distance ... ... ... 6 

Measuring distance of an inaccessible object 8 

Moon's distance determined from a fixed station ... ... 10 

Method depending on observations at distant stations ... 19 

Various results obtained by these methods ... ... 22 

Moon's distance determined from her motions ... ... 24 

Eesults by this method ... ... ... ... ... 25 

Moon's varying distance ... ... ... ... ... 27 

Moon's apparent diameter ... ... ... ... ... 28 

Her real diameter, surface, and volume 30 

Her varying apparent size ... ... .... ... ... 32 

The moon's mass determined in various ways ... ... 34 


The Moon's Motions. 

Introduction ... 

The law of gravitation, how established by Newton 
General consideration of a body's motion under gravity 
The "laws of motion 
The moon chiefly ruled by the sun ... 





"Why the moon does not leave the earth ... 73 

General consideration of the forces disturbing the moon ... 76 

Forces which pull the moon outwards {radial forces) ... 82 

How these forces vary during the year ... ... ... 84 

Resulting peculiarity of lunar motion called the annual 

equation ... ... ... ... ... ... ... 87 

A variation of this variation due to changing figure of earth's 

orbit (secular acceleration) ... ... ... ... ... ih. 

Forces which hasten or retard the moon (tangential forces) 91 

Resulting peculiarity of lunar motion called the variation ... 94 
Variation of this variation due to moon's varying distance 

from sun during the month (parallactic inequality) ... ih. 

Effect of tangential forces on the secular acceleration ... 97 

How the radial forces aJBFect the shape of the moon's path ... 99 
How they cause the place of the perigee to advance on the 

whole 100 

How the tangential forces aflfect the place of the perigee ... 109 

Why these forces hasten the advance of the perigee ... 113 

Actual motion of the perigee considered ... ... ... 124 

How the eccentricity is affected by the disturbing forces ... 127 

The evection 128 

Effect of the disturbing forces on the position of the plane of 

the moon's orbit 129 

General considerations ... ... ... ... ... 130 

Discussion of typical cases ... ... ... ... ... ih: 

Regression of the line of nodes 133 

Change of the inclination ... ... ... ... ... 1 36 

Conclusion ib. 


The Moon's Changes of Aspect, Rotation, Libration, &c. 


Sidereal month 

Tropical month 

Synodical month or lunation 






The moon's phases 141 

Varying position of the new moon's horns 142 

Rate at which the moon waxes and wanes ... ... ... 145 

Moon's varying path on the heavens ... ... ... ... 148 

Harvest moon and Hunter's moon 151 

Effects of the inclination of the moon's path to the ecliptic 158 

Effects of the eccentricity of the lunar orbit 164 

Effects of the motion of the nodes and perigee 166 

Moon's rotation 168 

Experiments illustrative of moon's rotation ... ... ... 171 

Lunar libration in latitude 173 

Libration in longitude .,. ... ... ... ... ... 177 

Effects of these librations combined 182 

Diurnal libration ... ... ... ... ... ... 191 

Extent of lunar surface swayed into and out of view by 

libration 193 

The moon's physical libration ... ... ... ... 199 

Shape of the moon's globe 200 


Study of the Moon's Surface. 


Views of Thales, Anaxagoras, the Pythagoreans, &c 

Observations of Galileo 

Work of Hevelius 

Eiccioli, Cassini, Schrbter, and Lohrman 

Beer and Madler 

Webb, Schmidt, Birt, &c 

Lunar photography 

The moon's total light compared with sunlight 

Variation of light with phase 

Moon's albedo J or whiteness ... ... ... 

Brightness of different parts of moon 

Changes of brightness ... ... ... ... ... ... ih. 

Conditions under which moon is studied 241 




Lunar features : 


. 244 

Crateriform mountains 

. 245 

Clefts or rills 

. 250 

Radiating streaks 

. 251 

Peculiarities of arrangement 

. 257 

Few signs of change 

. 258 

Search for signs of habitation ... 

. 259 

Varieties of colour 

. 262 

Changes due to sublunarian forces 

. 263 

Changes in Copernicus, Mersenius, Linne, &c 

. 266 

Heat emitted by moon 

. 272 

Researches by Lord Rosse 

. 278 


Lunar Celestial Phenomena. 

Question of a lunar atmosphere discussed 

Suggested existence of an atmosphere over the unseen henii 


Lunar scenery ... 

Aspect of the lunar heavens . . . 

Lunar seasons 

Motions of planets seen from moon, &c, 
Solar phenomena so seen 
Phases, &c., of the earth so seen 
A month in the moon 
Eclipse of the sun by the earth 




Condition of the Moon's Surface. 

Importance of studying the subject 
Moon's primal condition 
Moon exposed to meteoric rain 



Frankland's theory respecting lunar oceans 

Glacier action ... 

Mattieu Williams's theory of crater-formation (note) 
Mallet's theory of contraction of crust upon nucleus 
The radiations from Tycho, &c. 

Mr. Birt's photometric researches 

Detailed examination of the moon's surface required 
Eeal changes must be taking place 



Index to the Map of the Moon. 

Table I. Grey plains, usually called seas 383 

Table II. Craters, mountains, and other objects, num- 
bered as in Webb's Chart ib. 

Table III. The same alphabetically arranged 388 

Table IV. Lunar elements 393 



First Quarter, Feb. 27, 1871 Frontispiece 

Full Moon, May 14, 1870, 16h. 4m. 10s. 

Sid. Time Tofacep. 214 

Third Quarter, Sept. 16, 1870, Ih. 49m. Os. 

Sid. Time 230 


Plate I. Illustrating the measurement of the moon's 

distance Tofacep. 9 

II. Determining moon's distance by observations 

at Greenwich and Capetown 20 

III. The moon's orbit round the sun (showing 

eccentricity, &c.) 27 

IV. Illustrating Kepler's laws 59 

V. moon's motions ... ... ... 77 

VI. Showing the forces which perturb the moon 81 
VII. Illustrating the action of normal perturbing 

forces 99 

tangential 107 

motion of the perigee ... 115 

node ... . . 131 
advance of perigee and recession 









of nodes 

* These small photographs are not given in the subscribers' copies. 


Plate XII. Illustrating the moon's apparent motions, 

phases, &c To face p. 141 

XIII. the same, rotation, and libration 161 

XIV. libration in latitude and longitude 175 

XV. Libration-curves for various parts of moon's 

disc 177 

XVI. Showing the parts of moon affected by libra- 
tion 195 

XVII. Webb's chart of the moon (folio).* 

XVIII. Stereographic chart of the moon (folio).* 

To face each other at end of hooh. 
XIX. Bullialdus and neighbourhood, by Schmidt (4to).* 

To face y. 221 
XX. Portion of the moon's surface from a model by 

Nasmyth 250 

XXI. Lunar landscape, with " full " earth, &c.* 

XXII. with sun, "new" earth, &c.* 

To face each other between pp. 304 and 305 
Woodcut Copernicus (;Siecc?ii) 248 

* These plates will be found in the subscribers' folio volume, 
and are there given to avoid folding. 

Subscribers' copies contain, also, a photograph of the moon, first 
quarter, taken by Mr. Brothers on Dec. 27, 1865 ; and a series of 
smaU photographs showing the progress of the lunar eclipse of 
Oct. 4, 1865. These are intended to show what may be done with 
a refracting telescope (5 inches in aperture), not like Rutherfurd's, 
corrected for the chemical rays, but of the ordhiary construction. 
See pp. 230, 231. 


Plates XVII. and XVIII. should be so placed that when both 
are unfolded they will be side by side for comparison. 

Plates XXI. and XXII. should lie the same way, the top of each 
towards the left, so as to admit of being studied simultaneously. 


Plate VIII. fig. 30. - For " inwards," read " outwards." 
fig. 31 . For " outwards," read " inwards ." 
Plate X. fig. 39. The arrow near arc N'M' should be below the I, 
not above as shown, and the arrow near the arc NM should 
be above the I, instead of below. 
Plate Xni. fig. 55. Ml, near top, should be M2. 

For fig. 53a read 56a. 
Plate XV. fig. 77. The left-hand should be 0'. 
At p. 28, line 9 from bottom of page, for " more " read " less." 
p. 208, 3 for " that her, read " than 

that her." 
p. 349, 3 ,, top for " dusty," rmtZ " dusky." 




Although the sun must undoubtedly have been the 
first celestial object whose movements or aspect 
attracted the attention of men_, yet it can scarcely 
be questioned that the science of astronomy had its 
real origin in the study of the moon. Her compa- 
ratively rapid motion in her circuit around the earth 
afforded in very early ages a convenient measure of 
time. The month was^ of course, in the first place, 
a lunar time-measure. The i^^eek, the earliest divi- 
sion of time (except the day alone) of which we have 
any record, had also its origin, most probably, in 
the lunar motions. Then the changes in the moon^s 
appearance as she circles round the earth must have 
led men in very early times to recognize a distinction 
between the moon and all other celestial objects. 
While inquiring into the nature of these changes, 
and perhaps speculating on their cause, the first, 
students of the moon must have soon begun to 


recognize tlie fact that she traverses the stellar vault 
so as to be seen night after night among different 
star-groups. To the recognition of this circumstance 
must be ascribed the origin of astronomy properly so 
called. Until the varying position of the moon among 
the stars had been noticed, men must certainly have 
failed to notice the changes in the aspect of the stellar 
heavens night after night throughout the year. In 
examining the moon's motions among the stars, they 
must have been led to study the annual motion of 
the stellar sphere. Thence presently they must have 
learned to distinguish between the fixed stars and the 
planets. And gradually, as the study of the stars, 
the moon, and the planets continued, the fundamental 
problems of astronomy must have presented them- 
selves with increasing distinctness, to be for centuries 
the object of ingenious speculation, more or less based 
on the actual results of observation. 

It would be difficult to form just ideas respecting 
the order in which the various facts respecting the 
moon and her motions were ascertained by ancient 
astronomers. Indeed, it seems probable that among 
the various nations to whom the origin of astronomy 
has been attributed, the moon's changes of appear- 
ance and position were studied independently, the 
order of discovery not being necessarily alike in any. 
two cases. We are free, therefore, in considering 
the knowledge of the ancients respecting the moon, 
to choose that arrangement of the various facts which 
seems best suited to the requirements of the student. 


The first, as the most obvious peculiarity of the 
moon, is that continually varying aspect which has 
led men in all ages to select the lunar orb as the 
emblem of change. " The inconstant moon, that 
nightly changes in her circled orb,'^ must, in the first 
place, have appeared as a body capable of assuming 
really different shapes ; and it is far from unlikely 
that this apparent evidence of power, associated with 
the moon's rapid change of place among the stars, 
may have led to the earliest forms of Sabgeanism. 
Yet in very early times the true explanation of the 
peculiarity must have been obtained. The Chaldean 
astronomer undoubtedly recognized the moon as an 
opaque orb, shining only because reflecting the 
sun's light ; for otherwise we should be unable to 
explain the care with which they studied the moon's 
motions in connection with the recurrence of lunar 
and solar eclipses. Their famous cycle, the Saros 
(of which I shall have occasion to speak more parti- 
cularly farther on),- shows that they must have paid 
very close attention to the moon's movements for a 
long period before the Saros was determined, and for 
a much longer period before the cycle was made 
known to other astronomers of ancient times. More- 
over, as they recognized in the moon the occasion of 
solar eclipses, though they could see her waning as 
she approached the sun's place, and waxing from the 
finest crescent of light after passing him, it is clear 
that they must have understood that the lunar phases 
indicated no actual change of shape. Nor can we 
B 2 


imagine that reasoners so acute as the Chaldasan 
astronomers failed to recognize how all the phases 
could be explained by the varying amount of the 
moon^s illuminated hemisphere turned at different 
times towards the earth.* 

Quite early^ then, the moon must have been recog- 
nized as an opaque globe illuminated by the sun. It 
would be understood that only one half of her surface 
can be in light. And apart from the fact that the 
moon was early recognized as causing solar eclipses 
by coming between the earth and the sun, it would 
be understood by the fineness of her sickle when near 
the sun^s place on the celestial vault, that she travels 
in a path lying within the sun^s. That fine sickle of 
light shows that at such times the illuminated half 
is turned almost directly away from the earth ; and 
therefore the illuminating sun must at such times lie 
not far from the prolongation of a line carried from 
the earth^s centre to the moon^s. 

It is not improbable, indeed, that the acute Chal- 
dasans deduced similar inferences respecting the 
moon's nature from a careful study of her face ; 
for the features of the moon when horned or gibbous 

* It is remarkable, however, that Aratus, writing about 230 B.C., 
long after the time when the Chaldseans established their system of 
astronomy, refers to the lunar phases in a way which implies 
either ignorance or forgetfulness of their real cause ; for he 
speaks of the significance of the position in which the horns of the 
new moon are seen, regarding this position, though obviously a 
necessary consequence of the position of the sun and moon, as in 
itself a weather portent. 


obviously correspond with those presented by the full 
inoon_, in such sort that no one who considers the 
phenomenon attentively can doubt for a moment that 
the moon undergoes no real change when passing 
through her phases. It may also be imagined that 
the same astronomers who recognized the fact that 
Mercury is a planet, though he is never visible except 
in strong twilight, must have repeatedly observed 
that the whole orb of the moon can be seen when 
the bright part is a mere sickle of light. Nay, it is 
even possible that in the clear skies of ancient Chal- 
dsea * the chief lunar features might be discerned when 
the dark half of the moon is thus seen. 

The comparative nearness of the moon was pro- 
bably inferred very early from her rapid motion of 
revolution around the earth. Almost as soon as 
observers noticed that the celestial bodies have dif- 
ferent apparent motions, they must have learned that 
the moon^s daily change of place among the stars is 
much greater than that of any other orb in the 
heavens. It would seem almost, from the distinction 
drawn in Job between the sun and the moon, that for 
some time the moon was regarded as the only body 

* It is not very easy to determine what was the true site of the 
region spoken of in Judith (v. 6), ais the land of the Chaldseans. 
The verse here referred to shows clearly that the region was not in 
Mesopotamia. From astronomical considerations I have been led 
to suppose that the first Chaldean observers occupied a region 
extending from Mount Ararat northward as far as the Caucasian 
range. See Appendix A to " Saturn and its System," and the 
Introduction to my Gnomonic Star-atlas. 


which actually moves over the celestial vault ; for he 
says, '^ If I beheld the sun when it shined or the 
moon walking in brightness ^^ (Job xxxi. 26) ; and the 
recognition of the sun's annual circuit of the heavens 
most probably preceded the discovery of the motions 
of the planets. Be this, however, as it may, astro- 
nomers must quite early have ascertained that among 
the more conspicuous orbs not one travels so quickly 
over the celestial vault as the moon. Accordingly, 
we find that even in the very earliest ages of astro- 
nomy the moon was regarded as the orb which 
travels nearest to the earth; and in the system of 
Pythagoras, in which musical tones were supposed 
to be produced by the revolution of the spheres 
bearing the planets, we find the neate, or highest 
tone of the celestial harmonies, assigned to the 

Whether the Chaldaean astronomers ever ascertained 
the moon's distance observationally, is a question we 
have no means of answering satisfactorily. If they did, 
it is probable that the determination arose from the 
careful study of the moon's peculiarities of motion, 
undertaken with the object of rendering the prediction 
of eclipses more trustworthy. So far as is known, 
however, the first actual determination of the moon's 
distance (as compared with the dimensions of the 
earth's globe) must be ascribed to the astronomers 
of the Alexandrian school. Aristarchus of Samos 
(b.c. 280) had attempted to compare the distances of 
the sun and moon by a method of observation 


altogether inadequate to the requirements of that im- 
mensely difficult problem. * But he does not appear to 
have investigated the subject of the moon's distance. 
Somewhat more than a century and a quarter later, 
Hipparchus attacked both problems ; the first with 
no better success than had rewarded Aristarchus, but 
the second by a method which was probably very 
successful in his hands, though it is from his successor 
Ptolemy that we learn the actual results of observa- 
,tions applied according to the ideas of Hipparchus. 

It would appear that the scrutiny of the moon's 
motions, with the object of determining her path 
among the stars, and the exact laws according to 
which she traverses that path, led Hipparchus to 
attack the problem of determining the moon's distance. 
We know that his observations were so carefully pur- 
sued that he determined the eccentricity of the moon's 
path, and its inclination to the sun's annual path on 
the star-vault. It is also highly probable that he 
detected a certain peculiarity of the moon's motion, 
called the evectioUj which will be described further on. 
Whether this is so, or whether the discovery should 
be ascribed to Ptolemy, it is certain that the labours 
of Hipparchus could not have led to the results 
actually obtained, without his having noticed certain 
effects due to the relative nearness of the moon as 
compared with the other celestial bodies. The study 
of these effects probably enabled him to form a fair 
estimate of the moon's distance. 

* His method is described in my treatise on the Sun (p. 7). 


We have, however^ no record of tlie results actually 
obtained by Hipparcbus, and we must turn to tbe 
pages of tbe great work, tbe Almagest, written by 
Ptolemy about two centuries and a half later, for tbe 
first exact statement respecting tbe moon's distance, 
and tbe means used for determining it by tbe astrono- 
mers of old times. 

Tbe fundamental principle on wbicb tbe measure- 
ment of tbe distance of any inaccessible object de- 
pends, is a very simple one. If a base-line (A B, 
fig. 1, Plate I.) be measured, and tbe bearing of tbe in- 
accessible object from A and B (tbat is, tbe direction 
of tbe lines A 0, B 0, as compared witb tbe fine A B) 
be carefully estimated, tben tbe distances A C and B C 
can, under ordinary circumstances, be determined. 
For, in tbe triangle A B 0, we know tbe base-line 
A B, and tbe two base angles at A and B ; so tbat 
tbe triangle itself is completely determined. Tberefore, 
tbe ordinary formulse of trigonometrical calculation, 
or even a careful construction, will give us tbe sides 
A C and B 0. 

If in all sucb cases we could determine A B and tbe 
base angles at A and B exactly, we sbould know tbe 
exact lengtbs of A C and B 0. But even in ordinary 
cases, eacb observation must be to some extent, 
greater or less, inexact. Accordingly, tbe estimated 
distance of tbe object must be regarded as only an 
approximation to tbe trutb. Setting aside mistakes 
in tbe measurement of tbe base-line, mistakes in 
determining tbe angles at A and B will obviously 

I^J^A IJ'. J. 




^ ^'' 

'-;' ^V--c 


^i - 



Illustrating the Measurement of the Moon's Distance ,^i 



affect more or less seriously tlie estimate of either 
A or B 0. And a very brief consideration of tlie 
matter will show that the greater is the distance of C 
as compared with the base-line A B_, in other words, 
the smaller the angle 0, the more serious will be the 
effect of any error in the observation of the angles A 
and B. 

Now, the difficulty experienced by the astronomer 
in the application of this direct method to the deter- 
mination of the distances of celestial objects, consists 
chiefly in this : that his base-line must always be 
exceedingly small compared with the distance which 
he wishes to determine. It is, indeed, only in the 
case of the moon that the astronomer can apply this 
method with the least chance of success ; and even in 
her case the problem is by no means an easy one. 
We shall see presently that the distance of the moon 
exceeds the earth^s diameter in round numbers some 
thirty times. If the reader draw a figure, as in fig. 1, 
but so that each of the lines A C and B C is about 
sixty times as long as A B, he will see that the angle 
at C is exceedingly minute, insomuch that a very 
slight error in the determination of either of the base 
angles at A and B would lead to a serious error in the 
estimate of the distance of C, even supposing a full 
diameter of the earth could be taken as the base-line. 

Now, when we remember that the ancient astrono- 
mers were unable to undertake long voyages for the 
purpose of determining the moon's distance, and 
that, even though they could have set observers at 

10 THE MOON : 

widely distant stations^ they had not the requisite 
acquaintance with the geographical position of dif- 
ferent places to know what base-line they were making 
use of, it may appear surprising that Hipparchus or 
Ptolemy should have been able to form any satis- 
factory estimate of the moon^s distance. But Hip- 
parchus showed how the astronomer could deal with 
this problem without leaving his observatory. The 
eartPs daily rotation carries the astronomer's station 
each day round a vast circle, and he has but to notice 
the effect of this motion on the moon's position, to 
be enabled to form almost as satisfactory an estimate 
of her distance as by observations made at stations 
far apart. It is true that Hipparchus probably (and 
Ptolemy certainly) regarded the earth as fixed. But 
it is a matter of no importance (so far as the problem 
of determining the moon's distance is concerned) 
whether we regard the daily rotation of the moon 
with the celestial vault as due to the motion of the 
heavens themselves around the fixed globe of the 
earth, or as brought about by the rotation of the earth 
upon her axis. 

Let us now consider the features of this method 
attentively : 

In the first place, let us conceive the moon (fig. 2, 
Plate I.) to be at rest on the celestial equator, e E e' 
being the earth's equator, and P the earth's pole. 
Then, a place at e is carried by the diurnal rotation 
round the circle e E e'. If M e and M e touch the 
circle e E e' E', then, when the place is at e, the moon 


is seen on the horizon, due east ; and when the place 
has been carried to e , the moon is again on the 
horizon, but due west. When the place is at E, mid- 
way between c and e , the moon (under the imagined 
conditions) is immediately overhead. Thus, the moon 
rising due east, passes to the point overhead and 
onwards to the west, where she sets. But it is clear 
that the moon's apparent motion in passing across the 
sky, from the eastern to the western horizon, is not 
uniform, as seen from the globe E E'. The arc e E e' 
is obviously less than a semicircle ; in other words, 
the moon, under the imagined conditions, completes 
her course athwart the heavens a seeming half-circle 
in less than half a day, while she is below the 
horizon (completing the other seeming half of the 
circle) in more than half a day. 

But as some find a difficulty in forming a clear 
conception of the apparent motion of a body placed as 
M in fig. 2, while a point is carried round such a 
circle as E E', I will at this stage introduce a slight 
change in the method of considering the matter. It is 
of course obvious that the apparent motion of the 
moon is precisely the same as though the moon went 
round the earth, while the earth's globe remained 
at rest. Let us then suppose this to be the actual 
state of things. In fig. 3 the earth is supposed to be 
at rest, P being the pole, as in fig. 2 ; and the moon 
is supposed to be carried round uniformly about the 
earth's centre, in the direction shown by the arrow. 
Comparing figs. 2 and 3, the student will at once see 

12 THE MOON : 

liow one illustrates tlie real^ the other the apparent 
motion of the moon under the assumed conditions. 
Now, the line M^ M^ is the horizon line from east to 
west j Me the moon^s place of rising in the east ; M^^ 
her place of setting in the west. Then (always under 
the assumed conditions, which regard her as not moving 
on her orbital path) she appears to traverse the arc 
Me M M^ while above the horizon, and while below 
the horizon she traverses the arc M^ m M^. But, 
clearly M^ M M^y^ is less than a semicircle, and its 
difference from a semicircle depends entirely on the 
fact that the globe E E' has dimensions comparable 
with those of the circle M m ; in other words, that E P 
is comparable with M P. If we suppose the circle E E' 
drawn very much smaller, then the arc M^ M M^y be- 
comes very nearly a semicircle. If, on the other 
hand, we suppose the circle M-in drawn very much 
larger, then again the arc M^ M M-^^ becomes nearly a 
semicircle. So that, if observation shows the arc 
Me M M^ to differ appreciably from a semicircle, we 
have at once a means of determining the moon's 
distance as compared with the earth's radius. 

Suppose, for instance, that instead of taking twelve 
hours in passing from Me to M;^, the moon was ob- 
served to take only eleven hours, or 5| in passing 
from horizon to zenith, then we have only to draw a 
circle such as A B C D in fig. 4, Plate I. ; to divide the 
semicircle ABC into 12 parts, as shown, and to take 
B H, B H', each equal to h\ such parts ; then the line 
H H' cuts off for us P E, which represents the earth's 


radius, wliere PB represents the distance of tlie moon. 
Such a construction or, if preferred, the correspond- 
ing calculation would thus at once show what relation 
the moon^s distance bears to the eartVs diameter. 

It is obvious that although atmospheric refraction 
causes the moon^s apparent place, when she is near 
the horizon, to be somewhat higher than the place she 
would have if the atmosphere did not exist, yet this 
is a circumstance which the astronomer can take fully 
into account; since it is in his power, by observing 
the stars, to determine the exact value of atmospheric 
refraction on celestial bodies at different altitudes. 

This method of determining the moon^s distance is 
not the less available, that the moon is not at rest. 
Thus, suppose the moon to be travelling in the circle 
M m,, fig. 3 ; then, if the rate of the moon's motion be 
known, that is, the length of time in which the moon 
completes the circuit of the stars, the observer can 
apply to the moving moon precisely the same con- 
siderations which he would apply to the moon regarded 
as at rest. He would still be able to compare together 
the periods during which the moon is above and below 
the horizon, since her own motion would cause hoth 
these periods to be correspondingly affected. He 
would thus obtain the two unequal arcs M^ M M^y and 
M^y m Me (fig. 3), which would give him the cross 
line Me E M^, as before, and therefore the relative 
magnitude of E P and P M^. 

The actual problem is rendered somewhat less 
simple by the fact that the moon's motion does not 

14 THE MOON : 

take place in the circle M m, but in a path, inclined to 
that circle. But it is obviously in the power of mathe- 
matics to take into consideration all the ejffects due to 
the moon's real motion, and thus, as in the simpler 
case imagined, to deduce the relation between E P 
and Me P. 

But we may now look at the problem in a somewhat 
different light. Hitherto we have only considered 
the effect of the earth's size in causing' an apparent 
want of uniformity in the moon's rate of motion. We 
can see, however, from fig. 2, that what in reality 
happens is that the moon is not seen in the same 
direction from points on the earth's surface as from 
the centre of the earth; and that the apparent dis- 
placement is greater or less according as the moon is 
nearer to or farther from the horizon. If we suppose 
M to represent the moon's place when she is overhead, 
we see that she is seen in the same direction from E 
as from P. But when she is on the horizon at Mjj, 
she is seen as though ninety degrees from the point 
overhead ; whereas, as seen from P, she would be less 
than ninety degrees from that point : that is, she is 
seen from E lower down than she is in reality. In 
any intermediate position, as M', she would be seen 
lower down from E than from P; but not so much 
depressed as when she is near the horizon. 

But it is clear that this is equally true, wherever 
the station of the observer may be. The moon is 
always seen below the place she would occupy if she 
could be observed from the earth's centre, except 


wlien she is actually overliead ; and she is more de- 
pressed the nearer she is to the horizon. 

It follows that wheresoever the observer may be 
stationed on the earth, the moon cannot appear to 
move as she would if she could be watched from the 
centre of the earth. If M^ M3 M^ (fig. 5, Plate I.) 
represent her path as supposed to be seen from the 
centre of the earth, then the actual path she follows is 
as shown in the dotted line m^ m^ m^, her observed 
place being always vertically below her true place 
(for we may consider her place as supposed to be 
viewed from the earth's centre, her true place, since it 
is only as so viewed that her motions could show 
their true uniformity). This apparent displacement 
of the moon is called her parallax. 

Hence, for any observer not placed at a station 
where the moon rises actually to the zenith, it is not 
her total displacement when on the horizon, called her 
horizontal i^arallaXj which is to be compared with her 
true placing as she is seen on the zenith; but the 
former displacement is to be measured against the 
displacement which she shows when highest in the 
heavens. It is seen from fig. 6, Plate I., that when the 
moon rises high above the horizon this difference will 
be appreciable if the moon's horizontal parallax is ap- 
preciable. For let M represent the moon's place when 
she is 50 degrees above the horizon; then, as seen from 
P, she would lie in the direction P M ; but from E 
she is seen in the direction E M, which is the same 
as V m (drawing Pm parallel to E M). Thus the 

16 ' THE MOON : 

actual parallax, measured as an arc in the heavens, is 
represented by the arc M m. The horizontal parallax 
is represented by the arc Mh m,^, which is clearly 
greater than M m. If M' represents the moon^s place 
when she is 70 degrees above the horizon, then M' m', 
her parallax, is less again than M m.^ 

Now the moon's apparent diurnal path at any 
station on the earth would precisely resemble the 
apparent diurnal path of a star at the same distance 
from the pole, if it were not, first, for the moon^s 
actual motion amongst the stars, and secondly, for 
this effect, by which she is depressed below her true 
place more or less according as she is nearer to or 
farther from the horizon. The first circumstance could 
be taken into account so soon as the general course 
of the moon's motion came to be known. Her true 
path among the stars at any particular time could be 
ascertained. And then it would only remain to deter- 
mine how much she seemed to depart from that path 
when on the horizon, and again when high above it. 
This could be done by means of any contrivance 
which would enable the observer to follow the moon 

* The argument here relates to the actual construction of figures 
such as fig. 6 ; and the student should repeat the construction to 
satisfy himself on the point. The general mathematical determina- 
tion of the displacement is as follows : The arc Mh to^ is equal to 
E P (appreciably), and the arc M m is equal to a perpendicular 
from E on P m. Hence the parallax at M is to the horizontal 
parallax as the last-named perpendicular to E P, or as the sine of 
the angle M P Z. It follows that if the moon's horizontal parallax 
is H, her parallax when her true altitude is X, is H cos X. 


in the same way that the sun or a star can be followed, 
by means of a suitable pointer carried round the axis 
on which the celestial vault seems to rotate in what is 
called the diurnal motion; that is, around an axis 
directed to the true pole of the heavens. Such a 
pointer directed once upon a star would follow the 
star from rising to setting (neglecting the effects of 
atmospheric refraction) ; but directed on the moon, 
and corrected from time to time, so that the moon^s 
actual motions among the stars should be taken into 
account, the pointer would not follow the moon by a 
mere rotation around its polar axis. If pointed on 
the moon when she first rose above the horizon, it 
would be found to point below the moon when carried 
(round its axis) towards the place occupied by the moon 
when high above the horizon ; for it would have to be 
depressed by the full amount of the horizontal parallax 
when the moon was pn the horizon, and this depression 
would be too great when the moon was high above the 
horizon. In like manner, if the pointer were directed 
upon the moon when she was high above the horizon, 
it would be carried to a place above that occupied by 
the moon when setting beyond the western horizon. 

It was in this way that the moon's distance was 
first ascertained. The reader will recognize in the 
description just given the principle of the equatorial 
telescope, which, turning around a polar axis, follows 
a star by a single motion. But the astronomical prin- 
ciple of this instrument was understood and apphed 
long before the telescope itself was invented. Ptolemy, 


who is usually credited with, the invention of the equa- 
torially mounted pointer, was the first to apply the 
instrument to the determination of the moon^s dis- 
placement or parallax.* The result contrasts strikingly 
with the ill success which he and other ancient astrono- 
mers experienced when they attempted to apply this 
and other methods to the determination of the sun^s 
distance. He assigned 57' as the moon^s parallax 
when she is on the horizon, in other words, his 
observations led him to the conclusion that the 
angle EMuP (fig. 6, Plate I.) is one of 57', a value 
which would set the moon^s distance at almost 
exactly sixty times the earth's radius. We shall 
see presently that this is very close to the true value. f 
Other observations were made by this method; and 
it is probable that the value given for the lunar 

* A trace of this early application of the principle remains in 
the name parallactic instrument still sometimes given to the 
equatorial. The principle of the instrument is given in the 
Almagest, and the instrument, as made before the telescope was 
invented, was sometimes called Ptolemy's Rule. 

t Before this Aristarchus of Samos had set the moon's distance 
at two million stadia, which, according to Buchotte's estimate of the 
length of the Greek stadium, would be equal to about 230,000 miles. 
The method by which he deduced this result is not well known ; but 
it is believed to have been based on the consideration of the length 
of time occupied by the moon in passing from horizon to horizon ; 
in fact, it would seem to have been a modification of the method 
hypothetically considered in pp. 10 12. If so, it corresponded to 
a certain degree with the method he applied to determine the sun's 
distance. (See " The Sun," p. 25.) Hipparchus considered that the 
moon's distance lay between 62 and 72| times the radius of the 
earth. The above evaluation of Ptolemy is inferred from the 
numbers given at p. 211 of Prof. Grant's "History of Physical 


parallax in tlie Alphonsine Tables, viz. 58', was de- 
duced from a comparison of many such observations. 
This would give a distance somewhat exceeding 59 
times the earth's radius, or more exactly, with the pre- 
sent estimate of the earth's dimensions, 235,000 miles. 

Tycho Brahe, from his own observations, based on 
the same principle, found for the moon's horizontal 
parallax 61', corresponding to a distance somewhat 
less than 223,000 miles.* 

, But a more satisfactory method of determining the 
moon's distance is that which is based simply on the 
considerations discussed at pp. 8, 9, in other words, 
the method of observing the moon from two distant 
stations whose exact position on the earth's globe has 
been ascertained. 

Let us suppose, for convenience of illustration, that 
one station is the Greenwich Observatory, and the 
other the Observatory at the Cape of Good Hope. 

* Before passing from the consideration of the method of deter- 
mining the moon's distance by observations made at a single station, 
it may be mentioned that, as applied in later times, it depends on 
the moon's apparent displacement from her path, calculated for the 
earth's centre. Now since the moon's parallax always causes her 
to appear vertically below her true place, it is obvious that the ivhole 
of this displacement will operate to displace her from her calculated 
path, only when the part of the path which she is at the moment 
traversing is horizontal, in other words, when she is on the 
highest part of that path at the moment above the horizon. 
Although her actual parallax would then not be a maximum, 
it would act solely to shift her from her calculated path. 
According to the old astronomical systems, such occasions were 
held to be particularly favourable for lunar observations. The 
highest part of the moon's path was called its nonagesimal 
degree, a term also applied to the highest part of the elliptic. 
C 2 

20 THE MOON : 

These two stations are not on the same meridian, as 
will be seen from fig. 1 , Plate II., which shows Cape 
Town more than 1 8 of longitude east of Greenwich.* 
At present, however, we shall not take into account 
the difference of longitude. 

Let fig. 8, Plate II., represent a side view of the 
earth at night, when G-reenwich is at the place marked 
G. Let H 7t be a north and south horizontal line at 
Greenwich, G Z the vertical, G jp (parallel to the 
eartVs polar axis) the polar axis of the heavens ; and 
let us. suppose that the moon, when crossing the 
meridian, is seen in the direction G M ; then the 
angle _p G M is the moon's north polar distance. 

Again, let us suppose C to be the Cape Town 
Observatory, which has at the moment passed from 
the edge of the disc shown in fig. 8, by nearly 
1^ hours' rotation ; but let us for the moment neglect 
this, and suppose the station C to be at the edge of 
the disc. Let H' C li be the north and south horizontal 
line at C, C 7/ the vertical, C_p' (parallel to the earth's 
polar axis) the polar axis of the heavens (directed 
necessarily towards the south pole) ; and let us sup- 
pose that the moon, when crossing the meridian, is 
seen in the direction C M". Then, since the lines 
G M and C M' are both pointed towards the moon's 
centre, they are not parallel lines, but meet, when 
produced, at that point. 

Let fig. 9, Plate II., represent this state of things 

* This figure is reduced from one of the four summer pictures 
forming Plate VII. of my " Sun-views of the Earth." 

Determinirui JStoon's distance hy PLATE M 

Odsermiions at Greenwich &, Qi^eiown 


on a smaller scale, M being the moon, G Greenwich, 
and C the Cape of Good Hope ; then G C M is just 
such a triangle as we considered at page 8. The 
base-line G is of course known; and it is very easily- 
seen that the angles at G and C are known from 
the observations pictured in fig. 8.^ Thus M C and 
M G can be calculated. 

Such is the general nature of the method for deter- 
mining the moon's distance by observations made at 
different stations, and either simultaneously or so 
nearly simultaneously that the correction for the 
moon's motion in the interval can be readily made.t 

* The distance from Greenwich to Cape Town is not in question, 
but the distance between Greenwich and the point on the meri- 
dian of Greenwich ; for any effects due to the difference of longi- 
tude of Cape Town and Greenwich are readily taken into account 
astronomically. Now the distance C G is the chord of a known 
arc of a great circle of the earth, if we neglect the earth's ellipticity, 
or is a known chord of the elliptic section of the earth through her 
axis if we take the ellipticity into account (as we must of course 
do in exact measurement). Thus C G is known, and the angles 
O G C, C G, are equally known. Now the angle M G C is the 
sum of the angles M G H and H G C ; and of these M G H is 
the moon's observed meridian altitude at Greenwich, while H G C 
is the complement of the known angle G 0. Hence M G is 
known. In like manner M' C G is known. So that we have the 
base-line and the two base angles of the triangle M C G known, 
and therefore M C and M G can be calculated. In reality the 
angle M C G is about 1^ degrees. 

t If such an instrument as the equatorial were as trustworthy as 
a meridional instrument, it would be easy to make the observations 
simultaneously, determining the polar distances of the moon at 
Greenwich and Cape Town respectively. But as a matter of fact, 
it is absolutely necessary to observe the moon when she is on the 

22 THE MOON : 

One of the earliest series of observations directed 
to the determination of the moon^s distance was that 
undertaken by Lacaille when he visited the Cape of 
Good Hope in 1750. From a comparison of his re- 
sults with observations made in Europe, he deduced 
the value 57' 13''*1 for the moon^s mean equatorial 
horizontal parallax. This corresponds to a mean dis- 
tance of 238_,096 miles. But it is to be noticed that 
Lacaille was not acquainted with the true shape of the 
earth. He supposed the earth^s compression to be 
greater than it really is; in fact, he supposed the 
equatorial to exceed the polar diameter in the propor- 
tion of 201 to 199, whereas in reality the proportion is 
approximately 300 to 299 ; in other words, the com- 
pression is -3^0. If this correction is taken into ac- 
count, Lacaille's results give for the lunar parallax 
57' 4'''6, corresponding to a distance of 238,679 miles. 
Lalande, by comparing Lacaille's observations with, 
his own, made simultaneously at Berlin,* found for 
the lunar parallax the value 57' 3"* 7, corresponding 
to a distance of 238,749 miles. It will be noticed 

meridian. What then is done is to deduce from the observed north 
polar distance of the moon when on the meridian at Cape Town 
(or from the moon's place at that time, with respect to some 
known star) her position at the moment when she is on the 
meridian of Greenwich. 

" Lacaille was born on March 15, 1713, and Lalande on July 11, 
1732, so that Lalande was nineteen years younger than LacaUle, 
who was himself but a young man when he made his observations. 
In fact, Lalande was but nineteen years old when he was sent to 
Berlin for the purpose of observing the moon simultaneously with 
Lacaille at the Cape of Good Hope. 


that as Berlin is more than 13 degrees east of Grreen- 
wich, observations made on the moon when in the 
meridian, at Cape Town and at Berlin, are more 
nearly simultaneous than corresponding observations 
at Cape Town and Greenwich. 

Biirg, by comparing Lacaille's observations with 
those made at Grreenwich, deduced for the moon's 
parallax the value 57' V, corresponding to a distance 
of 238,937 miles. 

Henderson, the first who determined the distance 
of the celebrated star Alpha Centauri, made a series 
of lunar observations at the Cape of Good Hope in 
1832 and 1833, with very imperfect instrumental 
means. From a comparison of these observations 
with others made at Greenwich and Cambridge, he 
deduced 57' l''*8 for the value of the moon's parallax. 
The corresponding distance amounts to 238,881 miles. 

The Astronomer Royal, from a discussion of the 
whole series of Greenwich observations, deduced the 
value 57' 4"*94, corresponding to a distance of 238,656 

But probably the most accurate value is that which 
has been deduced by Professor Adams from a com- 
parison of Mr. Breen's observations at the Cape of 
Good Hope, with others made at Greenwich and 
Cambridge. Professor Adams deduces for the lunar 
parallax the value 57' 2"" 7, corresponding to a dis- 
tance of 238,818 miles. 

One other method of determining the moon's dis- 
tance remains to be mentioned. It cannot, however, 

24 THE MOON : 

be called a strictly independent metliod^, since it is 
based on the theory of gravity,, which could not have 
been established without an accurate determination of 
the moon's distance. 

In showing that the earth's attraction keeps the 
moon in her observed orbit^ Newton had to take into 
account the moon's distance. He reasoned that the 
earth's attraction reduced as the square of the dis- 
tance would be competent at the moon's distance to 
cause the observed deflection of the moon from the 
tangent to her path. He assumed the lunar parallax 
to be 57' 30'', corresponding to a distance of 237,000 
miles; and he found that the terrestrial attraction 
calculated for that distance corresponded very closely 
with the observed lunar motions, so closely as to 
leave no doubt of the truth of the theory he was 
dealing with. But now, when once the theory of 
gravity is admitted, we have in the observed lunar 
motions the means of forming an exact estimate of 
the earth's attraction at the moon's distance, and as 
we know her attraction at the earth's surface, we are 
enabled to infer the moon's distance. And in passing 
it may be observed that this process is not, as it 
might seem at a first view, mere arguing in a circle. 
Observation had already given a sufficiently accurate 
estimate of the moon's distance to supply an initial 
test of the theory that it is the earth's attraction 
reduced as the square of the distance which retains 
the moon in her orbit. This theory being accepted, 
and other tests applied, we may fairly reason back 


from it in such sort as to deduce the exact distance of 
the moon.* 

In this process, however, the mass of the moon 
would have to be taken into account. In fact, as will 
be seen in the next chapter, we must add the moon's 
mass to the earth's in considering the actual tendency 
of the moon towards the earth ; so that, if we know 
the moon's mass, the earth's size, and the moon's 
period, we can deduce the moon's distance.f 

Burckhardt applying this method, on the assumption 
that the moon's mass is ^ of the earth's, deduced 
the parallax 57' 0", corresponding to a distance of 
239,007 miles. Damoiseau, taking the moon's mass 
at tV of the earth's, deduced a parallax of 57' V, 
corresponding to a distance of 238,937 miles. Plana, 

"^ The case may be compared to the following : In determining 
the rotation period of Mars (see Appendix A to my " Essays on 
Astronomy "), I had certain dates, separated by long intervals, on 
which the planet presented a certain aspect. Now, knowing pretty 
accurately the rotation period, I could divide one of these long 
intervals by this pretty accurate period, to get the total number of 
rotations in the interval : I could be certain that I should not get 
a full rotation too many or too few, but only a small fraction of 
a rotation, which could very well be neglected. Then, having the 
number of rotations, I could reverse the process, dividing the 
interval by this number to obtain the rotation period more exactly, 
to obtain, in fact, a period which, used as a divisor instead of 
the former rougher determination, would leave no small fraction 
over or above. 

t The following is the treatment of the problem, on the assump- 
tion that the moon moves in a circle round the earth : 

Let P be the number of seconds in the moon's periodic time 
round the earth (the sidereal moilth) ; D, the distance of the moon 
in feet ; g, the measure of the force of gravity at the earth's surface 

26 THE MOON : 

assuming the moon^s mass to be g-Vj found for the 
mean lunar parallax the value 57' S'^'l, corresponding 
to a distance of 238,792 miles. 

We shall throughout the rest of this work assume 
that the moon's mean equatorial horizontal parallax is 
57' 2"- 7, and her distance, therefore, 238,818 miles, 
the earth's equatorial diameter being assumed equal 
to 7,925-8 miles. 

Now it follows from this that, as seen from the 
moon at her mean distance, the earth's equatorial 
radius subtends an angle of 57' 2"' 7; that is, the 
equatorial diameter of the earth covers on the heavens 
an arc of 1 54' 5"'4, as seen from the moon at her 
mean distance. If the moon's orbit were circular, 

(in other words, with the assumed units of time and space, gf = 32"2). 
Then the moon's velocity in her orbit 


~ p ; 

and the accelerating force of gravity exerted by the earth on the 



_ 1 /27rD\2 

~ D \ PA 

- ^R 


But the attraction g, first increased so as to take the moon's mass 
into account, and then reduced according to the law of the inverse 

where M is the earth's mass, m the moon's, and r the earth's radius. 
Hence, equating the expressions (i) and (ii) we find 

gr (M + m)P" r= { i 


J 1^/11 TjJII. 

In HmFi^ure, \^\y\iL,k\K\i' s?ieivJhe.Moon:s man^lea,if;,&jreate.s'fDzse. 


F/^ II: 17f(\ Moon 'y Or/)it . {-Marrnfum Eecfnfrmli/.) 



the earth^s equatorial diameter would always cover 
such an arc. But the moon traverses a path of con- 
siderable eccentricity. Its mean shape (for it varies 
in shape) is exhibited in fig. 10, Plate III._, where C is 
the centre of the orbit, E the earth, M the place of the 
moon when nearest to the earth, or in perigee, M' her 
place when farthest from the earth, or in apogee, m 
and -jji' her positions when she is at her mean distance 
(in other words, m m is the minor axis of the moon's 
, orbit). Thus E C is the linear eccentricity of the 
orbit.* E C is about the eighteenth part of C M, and 
is thus not at all an evanescent quantity even on the 
small scale of fig. 10. The distance E C is equal 
to about 13,113 miles. It will be observed, however, 
that though the eccentricitij of the orbit is shown in 
fig. 10, the ellipHcifyj that is the departure from the 
circular shape, is not indicated. In reality, it would 
not be discernible on the scale of fig. 10. t 

But the eccentricity of the moon's orbit is not 

* The true eccentricity is represented by the ratio of E C to 
E M ; that is, in the case of the lunar orbit, it is about ^V when the 
orbit is in its mean condition. When the orbit has its maximum 
eccentricity, the ratio rises to about -rV? ^^^ when the eccentricity 
is at its minimum, the value is about -^V- 

t By a well-known property of the ellipse, the distances E m 
and E m' are equal to C M and C M'. Hence C m is easily found. 
If, for convenience, we represent CM or Em by the number 18, 
E C will be represented by unity. Hence C m will be represented 
by -v/(18)2-l, or by ^323^ or by 17-9722. The semi-arcs C M 
and C m may be approximately represented by the numbers 1,800 
and 1,797 ; that is, by the numbers 600 and 599 ; or C m is less 
than C M by less than l-600th part of either. 

28 THE MOON : 

constant. Owing to the perturbations which the 
moon undergoes (as explained in the next chapter), 
her path changes in shape, the mean distance remain- 
ing throughout nearly constant. The shape of her 
path when it is most eccentric, as well as when it 
is least eccentric, would not differ appreciably from 
fig. 10, and therefore, so far as this relation is con- 
cerned, no new figure is required. But for another 
purpose, presently to be explained, it is convenient 
to have a picture exhibiting the moon's path around 
the earth when the eccentricity is a maximum. It is 
therefore shown in fig. 11, Plate II., the centre being 
at C and the earth at E', and M M' the moon's path. 
The point e shows the position occupied by the 
earth's centre when the eccentricity is a minimum. 
The distance E' is 15,760 miles, while e C is 10,510 
miles. Thus the difference, E' e, is 5,250 miles, or 
about two-thirds of the earth's diameter. Owing to the 
peculiarities of the lunar perturbations, however, these 
numbers are not to be strictly applied in dealing with 
the lunar orbit. In fact, her distance from the earth 
is somewhat more increased, owing to perturbations, 
than it is reduced ^when the maximum effects either 
way are compared. 

The apparent diameter of the moon when she is 
at her mean distance is found by telescopic observa- 
tion (at night) to be 31' 9'', or 1,869"' (when reduced 
to correspond to the distance of the earth's centre; 
or, approximately, when supposed to be made on the 
moon in the horizon). But this value is partly in- 


creased by the effects of irradiation. When the 
moon^s diameter is deduced from observations made 
during solar eclipses (at which time irradiation tends 
to reduce her apparent diameter_, because she is then 
seen as a dark body on a light ground), the value 
depends partly on the telescope employed. With 
instruments of average power it is about 30' 55'', or 
1,855". From a careful discussion of the occultations 
of stars by the moon, as observed at Greenwich and 
at Cambridge, the Astronomer Eoyal has inferred that 
the length of the moon^s mean apparent diameter is 
31' 5"'l, or l,865"'l.'^ This is the value assumed 
throughout the present work. (It is a useful aid to 
the memory to notice that the number of seconds of 
arc in this value gives the number of the year in 
which the Astronomer Royal announced his results.) 

* As inconvenience is often experienced from the absence of all 
explanations of estimates such as these, I here state how the above 
value has been inferred ; for I am unable to point to any passage in 
which the Astronomer Eoyal has distinctly stated it. In Madler's 
"Der Mond" it is stated, in 14, that Burckhardt assigns as the 
moon's semi-diameter 15' 31'''95. In the Monthly Notices of the 
Astronomical Society for 1864-65, the Astronomer Eoyal assigns 
2'' as the excess of the telescopic diameter of the moon over that 
inferred from stellar occultations ; and speaking of the eclipse of 
1833, he says that the observations of the moon gave 4"*2 as the 
correction on Burckhardt's semi-diameter, and 6"'8 as the correc- 
tion on the telescopic semi-diameter. It follows that the telescopic 
semi-diameter exceeds Burckhardt's by 2'''6, and therefore that 
Burckhardt's estimate is less than Airy's estimate from occulta- 
tions by 0"*6. Hence Airy's estimate from occultations (noivhere 
stated in his pajyer) must be 1 5' 32''"55, corresponding to an apparent 
mean diameter of 31' 5"-l. 

30 THE MOON : 

There is no apparent flattening of the lunar orb as 
seen from the earth ; the most careful measurement 
presents it as circular. Since the eartVs semi- 
diameter subtends from the moon an angle or arc of 
57' 2''- 7, or 3,422''-7, while the moon's diameter sub- 
tends from the earth an angle of l,86b'''l, it follows 
that the moon's diameter is less than the earth's 
radius (or 3^962 "9 miles) in the proportion of 18^651 
to 34,227. Thus it is readily calculated (by mere 
rule of three) that the moon's real diameter (or at 
least any diameter square to the line of sight from 
the earth) is 2jl59'6 miles. It chances that this is 
the exact value adopted by Madler_, though obtained 
by employing a dijSerent value of the lunar parallax, 
of the lunar apparent diameter, and lastly of the 
earth's real diameter. 

It follows that the earth's equatorial diameter exceeds 
the moon's in the proportion of about 3,670 to 1,000 ; 
or, if we represent the earth's equatorial diameter by 
10,000, then the moon's would be represented by 
2,725. Assuming the moon's shape to be globular, 
and the earth's compression ^^^, it follows that the 
earth's surface exceeds the moon's in the proportion 
of about 13,435 to 1,000; or, if we represent the 
earth's surface by 10,000, the moon's will be repre- 
sented by 744. Lastly, on the same assumption as to 
the moon's shape, the earth's volume exceeds the 
moon's in the proportion of about 49,263 to 1,000; 
or, if the earth's volume be represented by 10,000, 
the moon's will be represented by 209. 


Roughly, we may take tlie moon^s diameter as two- 
sevenths of the earth's, her surface as two twenty- 
sevenths, her volume as two ninety-ninths. Of these 
proportions, the most interesting is that between the 
moon's surface and the earth's ; for neither the dia- 
meter nor the volume of the moon is specially 
related to her condition as a globe comparable with 
our earth as respects those features which affect our 
own requirements. But the surface of the moon's 
globe obviously aifects her fitness, in one important 
respect, to be the abode of living creatures. Now the 
actual surface of the moon is rather more than two 
twenty-sevenths of the earth's, and the surface of the 
earth is about 196,870,000 square miles : hence the 
moon's surface is about 14,600,000 square miles. 
This is about the same as the area of Europe and 
Africa together (exclusive of the islands usually in- 
cluded with these continents). It is almost exactly 
equal to the areas of ISTorth and South America, 
exclusive of their islands. The portion of either 
hemisphere of the earth, lying on the polar side of 
latitude 58 23', is equal to the whole surface of the 
moon : that is, if E E', fig. 12, Plate I., represent the 
earth, P and P' being the poles, L I and L' V latitude 
parallels 58 23' north and south of the equator E E', 
then either of the spaces of which L P /, L' P' V are 
the visible halves, has an area equal to the moon's. 
The arctic and antarctic regions together exceed the 
moon in area in about the proportion of 10 to 9. 
Lastly, it may be noticed that, reckoning the Russian 

32 THE moon; 

empire (in Europe and Asia) at 7,900,000 square 
miles, and the Britisli dominions at 6,700^000, these 
two empires together are almost exactly equal in area 
to the whole surface of the moon : the part of the 
moon actually visible to us (taking her librations 
into account) is somewhat more extensive than the 
Russian empire, while the part totally concealed 
from ns is somewhat less extensive than the British 

It is important to notice that, under all circum- 
stances, whether the moon is at her mean distance, 
or nearer to or farther from the earth (in fact, 
whatever the size of her disc may be), the eartVs 
disc, as supposed to be seen at the moment from 
the moon, is nearly 13^ times larger. The actual 
proportion between the two discs is shown in fig. 7, 
Plate II. 

But the variation of the moon's apparent size, 
according to her varying distance, must also be 
carefully taken into account. It is much greater than 
is commonly supposed. The observed telescopic mean 
diameter of the moon is, as already stated, 31' 9'', 
while 31' 5"'l is taken as the true mean diameter, 
that is, the telescopic diameter reduced for the effects 
of irradiation. Now, the telescopic semi-diameter 
when the moon is at her nearest to the earth, that is 
to say not merely in perigee, but in perigee at a time 
when her orbit has its greatest eccentricity is found 
to be 33' 32"- 1, while, when the moon is farthest from 
the earth, the observed diameter is 29' 22"-9. These 


values reduced for the effects of irradiation_, give for 
the diameter, 

(1) When the moon is nearest to the earth, 33' 30*1" or 2010'1'' 

(2) at her mean distance, 31 5'1 or 1865'1 

(3) farthest from the earth, 29 20-9 or 1760'9 

It has been already mentioned (p. 28) that the mean 
distance is not the arithmetic mean between the 
greatest and least distance ; it necessarily follows that 
the mean apparent diameter is not the arithmetic mean 
between the greatest and least apparent diameters. 

Now, the apparent surface of the lunar disc varies, 
not as these diameters, but as the squares of these 
diameters. It is easily calculated that if the size of 
the lunar disc, when the moon is at her mean distance, 
is represented by the number 10,000, then, when she 
is nearest to the earth, her disc shows a surface of 
11,615; while, when she is farthest, the apparent 
surface is but 8,914. Or, if we call the surface of the 
moon^s disc when nearest to us 10,000, then, when 
she is farthest from us, the surface of her disc would 
be represented by the number 7,674. We may very 
nearly represent the apparent size of the moon's disc 
when she is nearest to us, and when she is farthest 
from us, by the numbers 4 and 3 ; in other words, 
when the moon is full and farthest from the earth, she 
gives only three-fourths of the amount of light which 
she gives when full and at her nearest to the earth. 
But there is a very convenient way of representing 
the relative dimensions of the moon's disc when she 
is at her nearest and farthest. It is very easily shown 


34 THE MOON : 

that if we describe circles Mju and M'ju' about E'as centre 
(fig. 11, Plate III.); and passing through the points 
M and M', then the circles M fi and M' fx' represent the 
dimensions of the lunar disc 'when the moon is at M' 
or M respectively. In like manner we could compare 
the dimensions of the lunar disc when the moon is in 
perigee and apogee, and the eccentricity has its least 
value {i.e. the earth as at e, fig. 11) ; or when the eccen- 
tricity has its mean value (the earth as at E, fig. 10) .* 

It remains only that we should consider the subject 
of the moon^s mass, that is, of the quantity of matter 
contained in her globe, whose volume or size is already 
known to us. 

There are four different ways in which the moon's 
mass may be determined; 

First, since we have already mentioned (and shall 
explain further in the next chapter) that the moon's 
motion under the earth's attraction is calculable when 
the size of the earth, the value of terrestrial gravity, 
and the moon's distance and mass are known, it 
follows that as the size of the earth, the earth's gravity, 

* This is a very convenient method of comparing the apparent 
dimensions of the same orb seen at diffprent distances. We take 
these distances, and with them describe circles ; then these circles 
represent the relative apparent dimensions, the largest, of course, 
corresponding to the appearance of the globe as seen at the least 
distance, and vice versd. Thus suppose that we .wish to compare 
the size of the sun as seen from two planets, which we may call, 
for convenience, P and P', and that we have a chart of orbits in- 
cluding the orbits of these planets ; then if the orbit of P represent 
the size of the sun as seen from P', the orbit of P' represents the 
size of the sun as seen from P. 


and tlie moon^s period are very accurately known, 
and as the moon's distance lias been determined by 
independent observations, her mass may be inferred 
by the consideration of her observed motions ; in fact, 
precisely as, in the method for determining the moon's 
distance, described at page 24, we infer the distance 
when the mass is known ; so, if the distance be inde- 
pendently determined, we can infer the mass.* And 
it is to be observed that although, if these two 
methods alone existed for determining the mass and 
distance, they* would leave both problems indeter- 
minate ; yet, as other methods exist, these two afford 
very useful tests of the accuracy of the results deduced 
by the other methods. 

Laplace, adopting the value 57'12'''03 for the lunar 
parallax, deduced for the moon's mass, by this method, 
the value ^^ ; the earth's mass being unity. 

Another method for determining the moon's mass 
is based on the theory of the tides. If the height 
of the tides at any place be observed carefully for a 
long period of time, and then the mean height of the 
spring tides be compared with the mean height of 
the neap tides, we can infer the relative efficiency 
of the sun and moon when acting together to raise 
the tidal wave, and when their actions are opposed. 

* It is easily seen that, on the assumptions made in the note at 
pp. 25, 26, the equations (i) and (ii) can either be used to give the 
result there stated, or to give the result 
w__ 47rD^ _ 
M ~ PV(7 ^ 

D 2 

36 THE MOON : 

The problem is indeed rendered difficult by theoretical 
and practical considerations of mucL. complexity. But 
presenting tlie problem roughly, we may say tbat, 
after careful attention to the observations, we obtain 
L + S and L S, where L is the lunar action and S 
the sun^s ; the first at spring tides, the second at 
neap tides. Now, the sum of these compound actions 
is 2L, and the difference 2S; so that we can infer L 
the lunar action, and S the solar action. These enable 
us to infer the relation between the moon^s mass and 
the sun^s. Newton was led by comparing the results 
of his theory with the observed height of the tides, to 
the conclusion that the moon's mass is ^ , the earth's 
being represented by unity. Laplace was led by the 
observation of the tides at Brest to the theory that 
the moon's mass is of the earth's. He considered, 
however, that this result, although less than Newton's, 
might still be considerably too large, since he judged 
that the height of the tides at Brest might be in- 
fluenced by several local circumstances. It seems 
obvious that this method cannot be susceptible of 
very great accuracy, since the figures of the ocean 
masses, as well with respect to their horizontal as to 
their vertical proportions, render the direct applica- 
tion of the theory of the tides impracticable. 

Another method depends on the circumstance that 
the earth circuits once in each lunation around the 
centre of gravity of the earth and moon. Owing to 
this circumstance, the earth is sometimes slightly 
in advance of, and sometimes slightly behind, her 


mean place in longitude. In fact we know tliat the 
moon J circling around the same centre of gravity^ but 
in a mucli wider orbit, is sometimes in advance of the 
earth and sometimes behind the earth, regarding 
these orbs as two planets severally pursuing their 
courses round the sun ; and if we look upon the earth's 
motion as representing very nearly the motion of a 
planet, at her distance and undisturbed by a satellite 
(which is not far from being the case), then wo see 
that the moon, owing to her motion in an orbit 
477,600 miles in diameter round the earth, is alter- 
nately 238,800 miles in advance of, and as many 
behind, her mean place in longitude. So that, since 
the earth circuits round the common centre of gravity 
of the two bodies, in a smaller orbit, she will be 
alternately in advance of and behind her mean place * 
by the radius of that orbit. Obviously the effect of 
this will be that the sun, round which the earth is 
thus moving, will seem to be alternately in advance 
of and behind the mean place due to his apparent 
annual motion round the heavens. His apparent 
place will obviously not be affected at all when the 
moon is on a line with the sun and earth, or in syzygy, 
as it is called (that is, when it is either new moon or 
full)j for then the earth's displacement is on the same 
line, and the only effect is that the sun appears either 
very slightly larger (when the moon is ^'"full-" and the 

* The mean place here referred to is that place which the earth 
would have if she were travelling alone round the sun, not, as 
is actually the case, under the perturbing influence of a satellite. 

38 THE MOON : 

earth most displaced towards tlie sun)^ or very slightly 
smaller (when the moon is " new ^^ and the earth most 
displaced from the sun) . Both effects would be quite 
inappreciable. But when the moon is at her first 
quarter^ the earth is displaced towards the side occu- 
pied by the moon at her third quarter ; that is_, she is 
at her maximum displacement in advance of her mean 
place, and the sun also appears accordingly at his 
maximum displacement in advance of his mean place 
in his apparent annual motion round the heavens. In 
like manner, when the moon is at her third quarter, 
the sun appears at his maximum displacement behind 
his mean place. It is easy to ascertain what the sun's 
displacement should be, on any given assumption 
as to the moon's mass. Suppose the moon's mass, 
for example, to be -gVth of the earth's, then the centre 
of gravity of the earth and moon lies eighty times 
farther from the moon's centre than from the earth's. 
Hence the distance of this centre of gravity from the 
earth is -g^st part of 238,818 miles, or 2,949 miles. 
Thus the sun may be displaced from his mean place 
by the angle which a line 2,949 miles long subtends at 
the earth's distance from the sun. Since the equa- 
torial diameter of the earth is 3,963 miles, this dis- 
placement of the sun is equal to about fths of the 
small arc called the solar parallax, or is rather more 
than Qi"'Q, if we assume 8'''9 to be the mean value of 
the solar parallax. This quantity is about ^vo^h part 
of the sun's apparent diameter. 

But obviously if the exact amount of the maximum 


displacement can be ascertained_, we can infer pre- 
cisely what proportion the distance of the earth's 
centre from the centre of gravity of the earth and 
moon bears to the earth's mean diameter. We shall 
have to make an assumption as to the value of the 
solar parallax (that is, in effect, as to the sun's dis- 
tance) ; but that is an element which has been deter- 
mined with a satisfactory degree of accuracy in many 
different ways. Hence the moon's mass can be deter- 
mined with a corresponding degree of accuracy, if 
only the observations of the sun's displacement are 
accurately made. 

From a great number of observations of the moon, 
Delambre deduced for the sun's maximum displace- 
ment (called the sun's parallactic inequality), the value 
7'''5. Hence Laplace deduced the value for the 
moon's mass. With the values at present adopted 
for the distances of the sun and moon, he would have 
deduced yV as the value of the moon's mass. 

In recent times the meridional observations of the 
sun have been so numerous and exact, that the means 
of determining the moon's mass by this method are 
much more satisfactory. Thus we can place very 
great rehance on Leverrier's estimate of the parallactic 
inequality, viz. 6'''50. Professor Newcomb, of America^ 
deduces from a yet wider range of observations the 
value 6''-52. These values lie so close together as to 
show that the observations on which they have been 
based sujSice for the very accurate determination of 
this quantity. 

40 THE MOON : 

Now tlie value of the moon^s mass which we should 
infer from the mean (6'^'51) of these two estimates, 
will depend on the yalue we assign to the solar 
parallax. If we estimate the mean equatorial hori- 
zontal solar parallax at 8''*91j it would follow that the 
distance of the centre of gravity of the earth and 
moon from the earth's centre is filths of the earth's 
equatorial semi-diameter, or ffrths of 3,963 miles; 
that is, about 2,895 miles. Thence it follows that 
the moon's mass is to the sum of the masses of the 
earth and moon as 2,895 to 238,818, or 

Moon's mass : Earth's mass : : 2895 : 235923 
:: 1 : 81-5* 

that is, the earth's mass exceeds the moon's 81^ 

In calculating the sun's distance from the solar 
parallactic inequality, Mr. Stone adopted ^ for the 
moon's mass. Leverrier adopted the value ^^ (origi- 
nally, owing to an error of calculation which Mr. 

* The actual relation may be given approximately thus : Let 
R be the earth's equatorial radius, D the moon's distance, P the 
sun's parallactic inequality, and n the sun's mean equatorial hori- 
zontal parallax, /x being the moon's mass when the earth's is repre- 
sented by unity ; then 

^ PR or;. PR 

fi+l nD nD-PR 

But the former form is more convenient for calculation. 

Leverrier takes as 0-016620 ; Newcomb adopts the value 

0-016461. The value resulting from the equatorial radius and the 
moon's distance adopted in the present work is 0-016593. 


Stone detected, Leverrier adopted the value g^) . Pro- 
fessor Newcomb adopted the value g-^. 

These values were deduced by an independent 
method, the last remaining to be described, and on 
the whole perhaps the most satisfactory. Owing to 
the attraction of the sun and moon on the bulging 
equatorial parts of the earth, the axis of the earth 
undergoes the disturbance called precession. Now 
this disturbance, whose period is about 25,868 years, 
depends on the inclination of the earth's equator-plane 
to lines drawn from the sun and moon. The portion 
due to the moon's action depends on the inclina- 
tion of the equator-plane to a line from the moon. 
Now of course this inclination varies during the 
moon's circuit of the earth, because she twice crosses 
the celestial equator in such a circuit, and at these 
times the moon's action vanishes. But these changes 
are comparatively unimportant so far as the progress 
of the displacement of the earth's axis is concerned, 
simply because the displacement during a month is 
exceedingly small. There is, however, a change which, 
having a much longer period, is clearly recognizable. 
The moon's orbit is inclined to the ecliptic by rather 
more than five degrees. If the orbit thus inclined 
had a constant position, its inclination to the earth's 
equator (assumed also to have a constant position, 
which is approximately the case), would also be constant. 
But we shall see in the next chapter that the direction 
of the line in which the moon's plane intersects the 
ecliptic, makes a complete revolution once in about 

42 THE MOON : 

18i years. Hence tlie inclination of tlie moon^s orbit 
to the equator is affected by an oscillation of rather 
more than five degrees on either side of the mean in- 
clination^ which is the same as that of the ecliptic to 
the equator,, or about 23^ degrees. Thus the inclina- 
tion passes in the course of rather more than 18^ 
years from about 18^ degrees to about 28| degrees, 
and thence to about 18^ degrees again. Obviously 
the lunar action varies accordingly; and, moreover, 
it is to be remembered that if the lunar action were 
alone in question, the pole of the equator would circle, 
not about the pole of the ecliptic, but about the pole 
of the moon's orbit-plane ; and as this pole is itself 
circling about the pole of the ecliptic in a period of 
rather more than 18^ years, it is readily seen that 
there will be a fluctuation in the motion of the pole of 
the heavens, having the same period. This fluctua- 
tion is necessarily small, because in 18| years the whole 
motion due to precession is small,* and this fluctua- 
tion is only a minute portion of the whole motion. 
It is found to amount in fact to about 9''*2, by which 
amount the pole of the heavens, and with it the appa- 
rent position of every star in the heavens, is at a 
maximum displaced from the mean position estimated 
for a perfectly uniform processional motion. Now, since 
this displacement (called nutation) is solely dependent 

* The 1360th part of the complete circuit made by the pole of 
the heavens round the pole of the ecliptic (less than 16' of a small 
circle of the heavens having an arc- radius of 23^ degrees), or 
about 6^' of arc. 


on the moon's mass, it follows that when its observed 
value is compared with the formula deduced by theory, 
a means of determining the moon's mass must neces- 
sarily be obtained. 

Laplace, adopting Maskelyne's value of the maxi- 
mum nutation, namely, 9''' 6, inferred for the moon's 
mass i (the earth's being regarded as unity). Pro- 
fessor Newcomb adopting 9'''223 for the lunar nuta- 
tion, and 50'^*378 for the annual luni-solar precession, 
'deduces the value ^^ . Leverrier with the same values 
deduces ^^ . Mr. Stone, in his latest calculation, with 
the same values, deduces for the moon's mass r^ . * 

In the present work we adopt -^ (or 0'01228) as 
the moon's mass, the earth's being regarded as unity. 
Taking the moon's volume as -^ (the earth's as 
unity), it follows that the moon's mass bears a smaller 
proportion to the earth's than her volume bears to the 
earth's volume, in the ratio of 4,926 to 8,140. Hence 
the moon's mean density must be less than the earth's 
in this ratio. So that if we express the earth's density 
by unity, the moon's will be expressed by 0'6052. If 
the earth's mean density be held to be 5*7 times that 
of water, the moon's mean density is rather less than 
3| times the density of water. 

Such are the main circumstances of that long pro- 
cess of research by which astronomers have been 
enabled to pass from the first simple notions sug- 

To these values may be added Lindenau's estimate ^, and 

the estimate obtained by MM. Peters and Schidlowski, 



gested by the moon^s aspect and movements^ to their 
present accurate knowledge of the distance^ diameter, 
surface, volume, and weight of this beautiful orb, 
the companion of our earth in her motion around 
the sun. 




Altoqethee the most important circumstance in what 
may be called the history of the moon^ is the part 
which she has played in assisting the progress of 
modern exact astronomy. It is not saying too much 
to assert that if the earth had had no satellite the law 
of gravitation would never have been discovered. Novj 
iudeed that the law has been established^ we can see 
amid the movements of the planets the clearest evi- 
dence respecting it, insomuch that if we could con- 
ceive all that has been learned respecting the moon 
blotted out of memory, and the moon herself annihi- 
lated, astronomers would yet be able to demonstrate 
the law of gravity in the most complete manner. But 
this circumstance is solely due to the wonderful per- 
fection to which observational astronomy on the one 
hand, and mathematical research on the other, have 
been brought, since the law of gravitation was estab- 
lished, and through the establishment of that law. It 
needs but little acquaintance with the history of 
Newton's great discovery, to see that only the over- 
whelming evidence he was able to adduce from the 


moon^s movements, could have enabled him to compel 
the scientific world to hearken to his reasoning, and 
to accept his conclusions. We can scarcely doubt that 
he himself would never have attacked the subject as 
he actually did, with the whole force of his stupendous 
intellect, had he not recognized in the moon's move- 
ments the means of at once testing and demonstrating 
the law of the universe. Had the evidence been one 
whit less striking, the attention of his contemporaries 
would soon have been diverted from his theories, 
which indeed could barely have risen above the level 
of speculations but for the lunar motions. Astronomy 
would never have attained its present position had 
this happened. It would have seemed vain to track 
the moon and the planets with continually increasing 
care, if there had been no prospect of explaining the 
peculiarities of motion exhibited by these bodies. 
Kepler had already done all that could be done to 
represent the planetary motions by empirical laws, 
the planetary perturbations could be explained in no 
such manner. The application of mathematical calcu- 
lations to the subject would have been simply useless; 
and there would have been nothing to suggest the 
invention of new modes of mathematical research, and 
therefore nothing to lead to those masterpieces of 
analysis by which Laplace and Lagrange, Euler and 
Clairaut, Adams, Airy, and Leverrier, have elucidated 
the motions of the heavenly bodies. 

The history of the progress of investigation by 
which Newton established the law of gravitation is 


full of interest. And although a high degree of 
mathematical training is requisite, in order fully to ap- 
prehend its significance, yet a good general idea of 
the subject may readily be obtained even by those who 
are not profoundly versed in mathematics. I propose 
to endeavour, in this place, to present the subject in a 
purely popular, yet exact manner. I wish the reader 
to see not merely how the law of gravity accounts for 
the more obvious features of the moon^s motion, but 
also how her peculiarities of motion her perturba- 
tions are explained by the law of attraction. On the 
one hand the Scylla of too great simplicity is to be 
avoided, lest the reader should be left with the im- 
pression that the evidence for the law of gravity is not 
so complete as it actually is ; on the other, the Oha- 
rybdis of complexity must be escaped from, lest the 
general reader be deterred' altogether from the in- 
vestigation of a subject which is not only extremely 
important but in reality full of interest. 

I invite the general student to notice, in the first 
instance, that the whole of the following line of 
argument must be attentively followed. If a single 
paragraph be omitted or slurred over, what follows 
will forthwith become perplexing. But I believe I 
can promise him that, with this sole 'proviso, he will 
meet with no difficulties of an important nature. On 
the other hand, should the more advanced student 
by chance peruse these pages, I invite him to con- 
sider that the account here presented is intended 
only as a sketch, and that if certain details are but 

48 THE moon\s motions. 

lightly treated, or omitted altogether, this has not 
been done without a purpose. 

It had been recognized long before Newton^s time 
that this globe on which we live possesses a power 
of drawing to itself objects left unsupported at any 
distance above the eartVs surface. It is, indeed, 
very common to find the recognition of this fact 
ascribed to Newton, who is popularly supposed to 
have asked himself ivliy a certain apple fell in his 
orchard. But the fact was thoroughly recognized long 
before his time. Galileo, Newton's great predecessor, 
had instituted a series of researches into the law of 
this terrestrial attraction. He had found that all 
bodies are equally affected by it, so far as his experi- 
mental inquiries extended; and he established the 
important law that the velocity communicated to fall- 
ing bodies by the earth's attraction increases uniformly 
with the time of falling ; so that whatever velocity is 
acquired at the end of one second, a twofold velocity 
is acquired at the end of the next, a triple velocity at 
the end of the third, and so on. 

In order to estimate the actual velocity which 
gravity communicates to falling bodies, Galileo caused 
bodies to descend slightly inclined planes. He showed 
that the action of gravity was diminished in the pro- 
portion which the height of the plane's summit bears 
to the sloped face; and by making the slope very 
slight, he caused the velocity acquired in any given 
short time to be correspondingly reduced. To reduce 
friction as much as possible, he mounted the descend- 


ing bodies on wheels, and made the inclined planes of 
hard substances perfectly polished. But other and 
better methods were devised ; and when Newton^s 
labours began, men of science were already familiar 
with the fact that a falling body, if unretarded by 
atmospheric resistance or other cause, passes in the 
first second over IGyV feet, and has acquired at the 
end of the second a velocity of 32^ feet per second ; 
by the end of the second second it has passed over 
64-1 feet in all, and has acquired a velocity of 64f feet 
per second ; at the end of the third it has passed over 
144y%, and has acquired a velocity of 96f feet per se- 
cond; and so on, the law being that the space fallen 
varies as the square of the number of elapsed seconds,* 
while the velocity varies as this number directly. 

So much, as I have said, was known before Newton 
began to inquire into the laws influencing the celestial 
bodies ; so that, if there is any truth in the story of 
the apple, Newton certainly did not inquire ivhy the 
apple fell to the earth. It is not impossible that on 
some occasion, when he was pondering over the 
motions of the celestial bodies, and perhaps think- 
ing of those inviting speculations by which Borelli, 
Kepler, and others had been led to regard the celes- 
tial motions as due to attraction, the fall of an apple 
may have suggested to Newton that terrestrial gravity 
afforded a clue which, rightly followed up, might lead 
to an explanation of the mystery. If the attraction of 

* The spaces traversed in successive seconds are proportional to 
the numbers 1, 3, 5, 7, &c. 



the sun rules tlie planets^ the attraction of the earth 
must rule the moon. What if the very force which dreiv 
the apple to the ground be the same ivhich Iceeps the dis- 
tant moon from passing aivay into space on a tangent 
to her actual orbit ! 

Whether the idea was suggested in this particular 
way or otherwise^ it is certain that in 1665, at the 
age of only 23 years, Newton was engaged in the in- 
quiry whether the earth raay not retain the moon 
in her orbit by the very same inherent virtue or attrac- 
tive energy whereby she draws bodies to her surface 
when they are left unsupported. 

In order to deal with this question, he required to 
know the law according to which the attractive force 
diminishes with distance. Assuming it to be identical 
in quality with the force by which the sun retains the 
several planets in their orbits, he had, in the observed 
motions of the planets, the means of determining the 
law very readily. The reasoning he actually em- 
ployed is not quite suited to these pages. I substitute 
the following, which the reader may if he please omit 
(passing to the next paragraph), but it is not difficult 
to grasp. Let us call the distance of a planet (the 
earth, suppose), unity or 1, its period 1, its velo- 
city 1 . Let the distance of a planet farther from the 
sun be called D ; then the third law of Kepler tells us 
that its period will be the square root of D x D x D, 
or will be D-ZD. But regarding the orbits as circles 
around the sun as centre, the circumference of the 
larger orbit will exceed that of the smaller in the pro- 


portion of D to 1 ; hence^ if the velocity of the outer 
planet were equal to that of the innor^ the period of 
the outer planet would be D. But it is greater^ being 
D v^D (that is, it is greater in the proportion of ^D 
to 1) j hence the velocity of the outer planet must be 
less, in the proportion of 1 to v D. Now the sun^s 
energy causes the direction of the earth^s motion to 
be changed through four right angles in the time 1 ; 
that of the outer planet being similarly deflected in 
'the time D v D ; and we know that a moving body 
is more easily deflected in exact proportion as its 
velocity is less; so that the outer planet, moving 
'/D times more slowly, ought to be deflected -/D 
times more quickly if the sun influenced it as much 
as he does the nearer one. Since the outer planet, 
instead of being deflected v D times more quickly, is 
deflected DvD times less quickly, the influence of 
the sun on the outer planet must be less than on the 
earth, \^ x D '/D times, that is, D x D (or D") times 
less. In other words, the attraction of the sun 
diminishes inversely as the square of the distance. 

Newton had ther efore only to determine whether 
the force continually deflecting the moon from the 
tangent to her path is equal in amount to the force 
of terrestrial gravity reduced in accordance with this 
law of inverse squares, in order to obtain at least a 
first test of the correctness of the theory which had 
suggested itself to his mind. Let us consider how 
this was to be done ; and in order that the account 
may agree as closely as possible with the actual his- 
E 2 


tory of the discovery, let us employ the elements ac- 
tually adopted by Newton at this stage of his labours. 
Newton adopted for the moon^s distance in terms 
of the earth^s radius a value very closely correspond- 
ing to that now in use. We may, for our present 
purpose, regard this estimate as placing the moon 
at a distance equal to sixty terrestrial radii. Thus 
the attraction of the earth is reduced at the moon's 
distance in the proportion of the square of sixty, or 
3,600, to unity. Now, let us suppose the moon's 
orbit circular, and let m m', fig. 13, Plate IV., be the 
arc traversed by the moon in a second around the 
earth at E {m m is of course much larger in proportion 
than the arc really traversed by the moon in a second), 
then when at m the moon's course was such, that if 
the earth had not attracted her, she would have been 
carried along the tangent line m t ; and if t be the 
place she would have reached in a second, then m t 
is equal to m m', and E t will pass almost exactly 
through the point mf. Thus t m', which represents 
the amount of fall towards the earth in one second, 
may be regarded as lying on the line t E.* Now 
in E is equal to m E, and therefore t m represents the 
difierence between the two sides m E and f E of the 

* In the account ordinarily given, t m' is taken as lying parallel 
to m E. This is also approximately true. As a matter of fact the 
point m' lies a little outside t E (that is on the side away from m) 
and a little within the parallel to m E, through t. But the angle 
t E m is exceedingly minute ; this angle as drawn represent- 
ing the moon's motion for about half a day instead of a single 
second of time. 


right-angled triangle m E t. Newton adopted the 
measure of the earth in vogue at the time, according 
to which a degree of arc on the equator was supposed 
equal in length to 60 miles,, or the eartVs equatorial 
circumference equal to 21,600 miles. This gave for the 
circumference of the moon^s orbit 1,296,000 miles, and 
for the moon^s motion in one second rather less than 
half a mile. Thus t m and ?n B are known, for m E 
is equal to thirty terrestrial diameters ; and thus it is 
easy to determine t E.* Now Newton found, that 
with the estimate he had adopted for the earth^s 
dimensions, t E exceeded m E by an amount which, 
increased 3,600-fold, only gave about 14 feet, instead 
of 16y\j feet, the actual fall in a second at the earth^s 

This discordance appeared to Newton to be too 
great to admit of being reconciled in any way with 
the theory he had conceived. If the deflection of the 
moon's path had given a result greater than the actual 
value of gravity, he could have explained the discre- 
pancy as due to the circumstance that the moon's 
own mass adds to the attraction between the earth 
and herself. But a less value was quite inexplicable. 
He therefore laid aside the investigation. 

Fourteen years later Newton's attention was again 
attracted to the subject, by a remark in a letter 
addressed to him by Dr. Hooke, to the effect that a 
body attracted by a force varying inversely as the 

* By Euc. I. 47 the square on E is equal to the squares on t M 
and M E. 


square of the distance,, would travel in an elliptic 
orbitj having the centre of force in one of the foci. 
I do not at present pause to explain this remark, 
which is indeed only introduced here to indicate the 
sequence of Newton's researches. It is to be noted 
that Hooke gave no proof of the truth of his remark ; 
nor was there anything in his letter to show that he 
had established the relation. He was not, indeed, 
endowed with such mathematical abilities as would 
have been needed (in his day) to master the problem 
in question. Newton, however, grappled with it at 
once, and before long the idea suggested by Hooke 
had been mathematically demonstrated by Newton. 
Yet, even in ascribing the idea to Hookers sug- 
gestion at this epoch, we must not forget that 
Newton, in the very circumstance that he had dis- 
cussed the moon's motion as possibly ruled by the 
earth's attraction, had implicitly entertained the idea 
now first explicitly enunciated by Hooke : for the 
moon does not move in a circle around the earth, but 
in an ellipse. 

In studying this particular problem, Newton's atten- 
tion was naturally drawn again to the long-abandoned 
theory that the earth's attraction governs the moon's 
motioDS. But he was still unable to remove the dis- 
crepancy which had foiled him in 1665. 

At length, however, in 1684, news reached him that 
Picard* had measured a meridional arc with great 

* Picard died at Paris in 1682, two years before the news of his 
labours had reached the ears of Newton. 

THE moon's motions. 55 

care^ and witii instrumental appliances superior to 
any wliich had been hitherto employed. The new 
estimate of the earth's dimensions differed consider- 
ably from the estimate employed by Newton before. 
Instead of a degree of arc at the equator being but 60 
miles in length, it now appeared that there are rather 
more than 69 miles in each degree. The effect of this 
change will be at once apparent. The earth's attrac- 
tive energy at the moon's distance remains unaffected, 
simply because the proportion of the moon's distance 
to the earth's diameter had alone been in question. 
Newton_, therefore, still estimated the earth's attrac- 
tion at the moon's distance as less than her attraction 
at her own surface, in the proportion of 1 to 3,600. 
But now all the real dimensions, as well of the earth 
as of the moon's orbit, were enlarged linearly in the 
proportion of 69 ^ to 60. Therefore the fall of the 
moon per second towards the earth, increased in the 
proportion of 3,600 to 1, was enlarged from rather 
less than 14 feet to rather more than 16 feet, agree- 
ing, therefore, quite as closely as could be expected 
with the observed fall of 16^^ feet per second in 
a body acted upon by gravity and starting from 

It is said that as Newton found his figures tending 
to the desired end, he was so agitated that he was 
compelled to ask a friend to complete the calculations. 
The story is probably apocryphal, because the calcu- 
lations actually required were of extreme simplicity. 
Yet if any circumstance could have rendered Newton 

56 THE moon's motions. 

unable to proceed with a few simple processes of 
multiplication and division, undoubtedly the great 
discovery which was now being revealed to him might 
have led to such a result. For he clearly recognized 
the fact that the interpretation of the moon's motions 
was not what was in reality in question, nor even the 
explanation of the movements of all the bodies of the 
solar system ; but that the law he was inquiring into 
must be, if once established, the law of the universe 

If we consider the position in which matters now 
stood, we shall see that in reality the law of gravita- 
tion had already been placed on a somewhat firm and 
stable basis. Newton had shown that the motions of 
the planets are conformable to the theory that the sun 
attracts each planet with a force inversely proportional 
to the square of the planet's distance. The motions of 
Jupiter's satellites (the only scheme known to Newton) 
agreed similarly with this law of attraction. And now 
he had shown that in the case of our own moon, the 
attraction exerted by the central body round which 
the moon moves, is related to the attraction exerted 
by this body, the earth, on objects at her surface, 
according to precisely the same law. Furthermore, it 
was known that all bodies are attracted in the same 
way by the earth, let their condition or elementary 
constitution be what it may. The inference seemed 
abundantly clear that the law of attraction, with 
effects proportional to the attracting masses, and in- 
versely proportional to the distances separating them, 


is the general law of matter, and prevails, as far as 
matter prevails, throughout the universe. 

But Newton was sensible that a law of this nature 
could not be established unless some special evi- 
dence, suited to attract the attention of scientific 
men to the subject, were adduced and insisted upon. 
The discovery must throw light on some facts hitherto 
unexplained, must in effect achieve some striking 
success, before men could be expected to look 
.favourably upon it. 

What Newton determined to do, then, was this. 
The law had been shown to accord with the general 
features of the lunar motions. But the moon^s motion 
is characterized by many peculiarities. At one time 
she takes a longer, at another a shorter time in 
circling around the earth, than that average period 
called the sidereal lunar month. At one time she is in 
advance of her mean place, calculated on the supposi- 
tion of a simple elliptic orbit ; at another time she is 
behind her mean place. The inclination of her path is 
variable, as is the position of its plane ; so also the 
eccentricity of her path and the position of her perigee 
are variable. Newton saw that if the law of gravita- 
tion be true, the moon's motion around the earth 
must necessarily be disturbed by the sun's attraction. 
If he could show that the peculiarities of the moon's 
motion vary in accordance with the varying effects 
of the sun's perturbing influence, and, still more, if he 
could show that the extent of the lunar perturbations 
corresponds with the actual amount of the sun's 


perturbing action_, the law of gravitation would be 
established in a manner there could be no disputing. 

In presenting so much of the history of this inquiry 
as is necessary for my present purpose, it is necessary, 
in the first instance, to show generally how the motion 
of a body in an elliptic orbit accords with the action 
of a force like gravity. Absolute proof of the fact 
requires in the learner an amount of mathematical 
knowledge, which the general reader cannot be sup- 
posed to possess. But the difficulties which at a first 
view surround the idea of elliptic motion, or of motion 
in any non-circular orbit, described under attractive 
influences, can be removed without dealing with 
mathematical considerations. I think the most salient 
difficulties are the following : 

Suppose AB ah, fig. 14, Plate IV., to be an elliptic 
path described about an attracting body S, placed 
at one focus of the ellipse, then the learner finds 
some difficulty in understanding how the change of 
distance from the small distance S A to the great 
distance S a, and, vice versa, can proceed in regular 
alternation. Because, if the attracting force, greatly 
reduced at the distance S a, can nevertheless compel 
the body to approach from that distance until its 
distance is reduced to S A, how much more, it would 
seem, should the much greater attraction exerted at 
this reduced distance S A, continue to cause the ap- 
proach of the body, until finally the latter is brought 
to rest at S. Or again, if when the attracting orb is 
exerting its greatest influence on the moving body at 





F//jf /4 

lUii^'ilra/iiuj the rnulio)i of/f dodu nrotuid a/i allr/KjHi 


A, this body is still able to move in sucli a way as 
continually to increase its distance until it is as far off 
as a from S^ how much more is it to be expected that, 
having reached this distance, where the sun^s force is 
so greatly reduced, the body should be able yet farther 
to increase its distance, and so to travel for ever away 
from S. 

And I think that another difficulty, which is very 
commonly experienced, is this : The curve A B a & is 
quite symmetrical, both as respects the line A a and 
the line B h. Thus the part near A is exactly like the 
part near a ; yet these perfectly similar parts are de- 
scribed under quite dissimilar circumstances, the 
attraction on the body being different, the velocity of 
the body being different, and all the circumstances in 
fine at a maximum of dissimilarity. Nay, the very 
circumstance that a symmetrical orbit should be de- 
scribed about an eccentrically-placed point, seems at 
a first view inexplicable. 

Let S, fig. 14, Plate IV., be the attracting orb around 
which a body is moving in the elliptical orbit A.'^ah ; 
and let us consider the motion of the moving body 
from the point A, where the velocity is greatest. At 
this point the velocity is greater than that with which 
a body would describe the circle A L K around S : the 
tendency to travel on the tangent line A Y is therefore 
stronger than in the case of such a body. Thus an 
intermediate course, A P, is pursued, the sun's in- 
fluence deflecting the moving body from the tangent- 
line AY; but not being strong enough to deflect it 


into the circular course AL K. Now^ tlie distance of 
tlie body from S is increasing throughout this process, 
and this amounts to saying that a tangent-hne, as 
P T, makes an obtuse angle with the line S P drawn 
to the body at the moment. But this being so, it is 
obvious from the figure that the orb at S must exert 
a retarding influence. At A there was no retardation 
(for the moment), because the pull was square to the 
body's course ; but so soon as the body's distance be- 
gins to increase, the pull is partly backwards (as at P) 
with reference to the body's motion ; and thus there 
is retardation. Now two opposing influences are at 
work when the body is in such a position as P : one, 
the tendency of the body to move in the direction 
P T, tends to enlarge the angle S P T ; the other , 
the pull of the orb at S, tends to reduce this angle. 
So long as the velocity exceeds a certain value, the 
former influence prevails. But the velocity is being 
continually reduced ; and though the distance of the 
body is increasing, and therefore the pull from 8 
diminishing, yet the power of S to deflect the body 
does not diminish so rapidly as the absolute power of 
S on the body, for deflection becomes so much the 
easier as the velocity of the body is reduced. At 
length, when at B, the body has reached a position 
where the two forces counterbalance each other in 
this respect, the angle S B Z between the line of the 
body's motion and the line from S having here its 
maximum value. At this point the body is travelling 
on a course square to the direction it had had when 

THE moon's motions. 61 

at A.* And let this be noted as to the present con- 
dition of the body. It has increased its distance from 
Sj and has thus far asserted, in a sense_, the power 
inherent in it when at A_, by virtue of its high velocity 
there ; it has also increased its angular rate of escape, 
the angle S B Z exceeding the angle SAY. But in 
effecting this it has sacrificed a portion of its velocity, 
the influence of the orb at S having acted retardingly 
throughout the whole of this portion of the body's 
course ; and, as a matter of fact, the velocity at B is 
less than the velocity at A, in the proportion that A S 
is less than B C. 

As the body passes onwards from B, the sun's 
action continually reduces the angle corresponding to 
S P T, S B Z, continually reducing also the velocity of 
the body so long as this angle remains obtuse. This 
process is in this resjpect the reverse of the process 
passed through as the body moved from A to B, and 
ends in the restoration of the rectangular thwart 
motion when the body has arrived at , directly oppo- 
site to A. Yet the part Bp a of the body's course is 
described under circumstances wholly different from 
those operating while the body was moving from A 
to B. The time from A to B is much less than the 
time from B to a, in the same proportion, in fact, 

* It will be remembered that in the above paragraph explana- 
tion of what actually happens, and not a proof that it must hap- 
pen, is attempted. To show that when the angle corresponding to 
S P T ceases to increase^ the body must be travelling on a course 
at right angles to A Y, is impossible without introducing mathe- 
matical considerations much more fully than is proper in this place. 

62 THE moon's motions. 

that the area A S B is less than the area B S a ; and 
as the body moves more sluggishly from B to a, so 
also it is more sluggishly retarded by the orb at S. 
But it is precisely because of these opposite differences 
that the course of the body from B to a resembles 
in shape the course from A to B. To show that this 
is in accordance with the facts, that is, to explain 
the relation without undertaking to prove that it must 
hold, let us consider the state of the body at p, a 
point symmetrically placed with respect to P (that is, 
as far from B C, a C, as P is from B C, C A). At p the 
course of the body is for the moment in direction 
p t, and by the properties of the ellipse the angle Sjp ^ 
is equal to the angle S P T ; so that as far as direction 
is concerned the retarding influence of the orb at S is 
as effective on the body when at p as when at P. In 
magnitude, however, the pull is less in the proportion 
of the square of S P to the square of S p. But to make 
up for this, exactly, the velocity at p is less than the 
velocity at P, in the same proportion that S P is less 
than Sp.'^ This deficiency acts doubly (or, as it were, 
squares itself) : for the reduced velocity causes the 

* The velocities in an elliptic orbit are not generally proportional 
inversely to the distances from the attracting centre, but they are in 
the case of two such positions as p and P. In reality, because of 
the equal description of areas the velocities at different points 
are inversely proportional to the perpendiculars from the point S to 
the tangent through the respective points. But if tangents were 
drawn from S to P T and p t, these perpendiculars would clearly 
be proportional to S P and S p, simply because S P and S p are 
equally inclined to P T and p t. 


body to remain proportionately longer under the in- 
fluence of the body at S while describing any given 
small arc at p than when describing a corresponding 
one at P ; and it also causes the deflecting influence 
of the body at S to be proportionately more eff'ective. 
Thus the actual curvature of the path at p is exactly 
equal to the curvature at P. 

At a, then, the curvature is the same as at A, or 
the path lies luithin a circular arc, a.s la Ic, about S as 
centre. The distance of the body from S begins now 
therefore to diminish. Nor is it difficult to see that 
the course now pursued by the body must be the 
exact counterpart of that already traversed_, only pur- 
sued in a reverse order ; for all the circumstances are 
symmetrically reversed, so to speak. The distance of 
the body diminishing, the course of the body must be 
inclined at an acute angle to the line from S, and the 
influence of S must therefore act to accelerate the 
motion of the body. Thus when the body is at p\ 
(as far from S as p is), its course lies for the moment 
in the direction p' f , and the pull of the orb at S, 
acting in direction j/ S, must needs accelerate the 
body^s motion. Also, as the body starts from a with 
the same velocity that it had when it reached a, and 
moving at the same angle with S a, it is clear that the 
reduction of its velocity and distance from S, after it 
has passed a, must be affected in a manner precisely 
corresponding (point for point of its course) to the 
increase of its velocity and distance from S before it 
reached a. Up to the point h (corresponding to the 


point B) the angle between the course of the body 
and the line drawn from S continues to diminish^ and 
at h this angle has its minimum value^ Sh z. At 
this point h, the body has recovered a portion of the 
velocity it had lost^ but its distance has diminished, 
and its course is now directed as nearly towards the 
body at S as it can possibly be. After passing h the 
continual access of velocity, owing to the sun^s at- 
tracting force, causes the body to travel on a course 
inclined at a continually increasing angle to the line 
from S, but the distance of the body continues to 
diminish, until at A, where the angle between the 
course of the body and the line from S is again a 
right angle, the distance is reduced to the minimum 
value S A, as at first. All the circumstances are now 
the same as when the motion began. 

It is to be noticed o-f the above explanation, that 
though it does not prove that an ellipse must be 
described, it shows that the description of an ellipse 
corresponds with the circumstances of the case, that, 
in fact, in each quadrant of the ellipse forces tending 
to produce motion in a curve of such a shape, are in 
operation. This is all that can be done by way of 
popularly explaining a proposition whose inherent 
difficulty is such that eminent mathematicians like 
Wren and Halley failed to solve it.* But the above 

* It has been objected even that Newton's demonstration is 
imperfect inasmuch as it only shows that the curvature at any 
point of a conic section corresponds with that due to the law of 
force according to the inverse squares of the distances. But taken 


explanation removes in reality the real difficulties 
experienced by the learner ; for it shows that the equal 
curvatures at corresponding points,, P^ ;p, p', and P', 
in the four quadrants A B, B a^ ah, and h A, is a 
relation according with the amount of force exerted 
by the orb at S on the moving body at these four 
points. This has been already indicated as respects 
the points P and p, and holds in like manner as 
respects the points p and P' ; while it needs no de- 
monstration to show that at p' the curvature must be 
the same as at p, since the velocities at these points 
are equal, the forces on the moving body also equal, 
and the retarding action at p precisely accordant with 
the accelerating action at p', so far as the production 
of curvature is concerned; and lastly, it follows in 
like manner that the curvature at P' is equal to the 
curvature at P. 

Before passing from this investigation of elliptic 
motion, it may be well to notice in what respect the 
points A and a, B and h are critical points of the 
body's motion : 

(i.) At A the velocity is at a maximum, the dis- 
tance at a minimum, and the direction of the body's 
motion has a mean value, being at right angles to the 
line from S. 

(ii.) At a the velocity is at a minimum, the distance 
at a maximum, and the direction of the body's motion 

in its proper place, and in conjunction with what precedes and 
follows, the demonstration is in reality complete 

has again a mean value^ being again at right angles to 
the line from S. 

(iii.) At B the direction of the body^s motion is 
inclined at a maximum angle to the line from S^ the 
distance has its mean value B S^ being the arithme- 
tical mean between A S and S a ; and the velocity has 
what may be entitled its mean value, being the geome- 
trical mean between the velocities at a and A. 

(iv.) At h the same conditions prevail as respects 
distance and velocity as at B, but the direction of the 
body^s motion is inclined at a minimum angle to the 
line from S."^ 

The relation which we have been considering cor- 
responds to the first law which Kepler recognized in 
the planetary motions; viz., that each planet travels 
in an ellipse, the sun being situated at one focus of 
the curve. This law is not strictly true for the planets, 
or indeed for any known case in nature, since no orb 
is free to revolve around another quite independently of 
extraneous attractions. The law is, however, approx- 
imately true when any orb is subject almost wholly to 
the attraction of a single body ; or else, though sub- 

* Since any point of the orbit may be regarded as a starting- 
point, we notice that the same shaped curve is described whether a 
body is projected as at B on a course making the obtuse angle S B Z 
with the line from S, or with the same velocity from the equidis- 
tant point b, on a course making the acute angle S t Z with the 
line from S. The more general proposition also holds, that in 
whatever direction a body be propelled from a given point and with 
a given velocity, its orbit will have a major axis of constant 

THE moon's motions. 67 

ject to other attractions^ yet so shares these attractions 
with another orb that in its motions round this orb 
it may be regarded as almost wholly under its in- 
fluence. For instance, the law approximately holds 
in the case of a planet's motion around the sun : and 
it is also true of the motion of the moon around the 
earth, though the moon is chiefly under the sun's 
influence; for the earth and moon are both swayed 
almost equally by the sun. 

The second law of Kepler, as applied to the moon, 
also concerns us here very importantly. It was thus 
presented by Kepler : The line drawn from the sun 
to a planet sweeps over equal areas in equal times. 

Thus if S (fig. 15, Plate lY.) be the centre around 
which a body is revolving in .the path AB ah under 
the influence of gravity, and if in any given equal 
intervals of time the body passes from A to 1, thence 
to 2, thence to 3, and so on, then the spaces A S 1, 
1 S 2, 2 S 3, 3 S 4, and so on, are equal in area. For 
example, if the path were carefully drawn on paper 
according to true scale, then, if the spaces just named 
were cut out and carefully weighed, it would be found 
that they were exactly equal in lueigld. 

The third law of Kepler does not directly concern 
us here, because it deals with the relation between 
the mean distances and periods of different bodies 
travelling around one and the same centre. Never- 
theless, as the moon's motions are subject to changes 
of velocity, direction, and so on, while the attraction 
actually drawing the moon towards the centre of the 
P 2 

68 THE moon's motions. 

earth, is variable (because partly depending on the 
sun, and therefore on the moon's position), it is de- 
sirable to have clear ideas at the outset as to the 
effects of such changes. The third law of Kepler 
bears directly on this subject. It is as follows : 

The squares of the periods in which the planets 
travel around the sun vary as the cubes of the mean 

This law would be strictly true if the planets were 
infinitely minute compared with the sun; but the masses 
of the planets, though very small, bear yet definite 
relations to the sun, and, as a matter of fact, instead of 
considering each planet as swayed by the sun's mass, 
we must regard each as though swayed by the sum of 
its own mass and the sun's, supposed to be gathered 
at the sun's centre. This at least is a sufficient rule 
as regards the period of a planet and the dimensions 
of its orbit with respect to the sun ; though of course 
to determine the actual orbit around the common 
centre of gravity, we should have to take into account 
the actual disposal of the masses forming this sum. 
So that, in effect, to obtain the exact law for the 
periods and mean distances of the planets, we have to 
regard them, not as bodies circling around the same 
centre, but as so many different bodies revolving 

* More exactly thus : Fixed units of time and space being 
chosen, the square of the number expressing the periodic time of 
a planet bears a constant ratio to the cube of the number express- 
ing the mean distance of the planet. 

The mean distance is equal to half the major axis of the orbit. 


around centres slightly differing in attractive energy ; 
Jupiter, for instance, around a centre equal in mass to 
Jupiter and the sun ; Saturn round a centre equal in 
mass to Saturn and the sun ; and so on. The result of 
this consideration is that, instead of finding the frac- 
tion [=ii!!!^!H (.Qnstant for the solar system, we find 

{period)2 >' ' 

that this fraction calculated for the different planets 
(1) Mercury, (2) Venus, (3) Earth, and so on, gives 
results respectively proportional to (1) the sun^s mass 
added to Mercury's, (2) the sun's mass added to 
Yenus's, (3j the sun's mass added to the earth's, and 
so on.* 

* The law thus interpreted is applicable to all cases where differ- 
ent bodies revolve around a common centre. But it also admits 
of being generalized for different bodies travelling round different 
centres. Thus extended, it runs as foUows : 

If a body of mass m revolves round a centre of mass M in 
time P, and at a mean distance D, and another body of mass m' 
revolves round another centre of mass M' in time P', and at a 
mean distance D', then 

W- _ D'3 

P2 (M +^) ~ P'2 (M' + mO 

This general law, almost as simple, be it observed, as Kepler's third 
law, is extremely important. It may be regarded as the fundamental 
law of the celestial motions. It presents the influence of gravity as 
a bond associating the motions of all the orbs in the universe, 
whether of double suns around each other, or of primary planets 
around suns, or of secondary planets around their primaries. It is 
a law absolutely universal (so far as is known), and strictly exact, 
excepting in so far as perturbations come into operation to affect 
it ; and as perturbations have very little effect on mean periods 
of revolution, the exactness of the law is scarcely affected in this 
way. It is a wonderful thought that we can by means of such a 
law associate the motions of bodies, which to ordinary apprehen- 


Let us now pass on to tlie subject of tlie moon^s 
perturbations caused by the sun's attraction. 

Here^ in the first place, I may mention a fact which 
will perhaps seem surprising to many. Though the 
sun's disturbing influence on the moon is such that 
the moon's course around the earth is not very dif- 
ferent in any single revolution from that which she 
would have if the sun's attraction had no existence ; 
yet the sun actually exerts a far more powerful in- 
fluence on the moon than the earth does. As we 
shall have to consider the relation between the two 
forces, we may as well proceed at once to prove this 
excess of power on the sun's part. 

The law of gravitation enables us at once to com- 

sion have nothing in common ; that, for instance, such a relation as 
this can be affirmed : 

, FMoon's mean distancelS P Mean distance between "13 

L from earth. J [ components of a Centauri. J 

t Moon's mean l:i P Sum of moon's "1 f Their period T2 T 8un> of ~j 
period. J [_ma83 and earth's. J Lof revolution. J |_their masses. J 

It will be observed how the law enables us at once to compare the 
sums of the masses, when we know the mean distances and periods. 
For it may be written 

M + m ^ D^F" 

M' + m' 1)'-' Y' 

Also where m and m! are both small, compared with M and M' 
respectively, the law becomes simplified into 

M _ WV^ 

This law is in effect applied, in what immediately follows in the 
main text, to the determination of the moon's mass. It is there 
also independently established, at least in the case of circular 

THE moon's motions. 71 

pare tlie sun's mass with the earth's. For precisely 
as we have been able to show that under the influence 
of terrestrial gravity the moon, at her distance, should . 
follow such a path as she actually traverses, so we can 
determine how much a body should be deflected per 
second at the earth's distance from the sun, if his 
mass were equal to the earth's ; and by comparing this 
amount with the actual deflection, we can compare the 
sun's mass with the earth's. 

Or we may proceed in this way : 

The earth, at a distance of 238,800 miles from the 
moon, has power to deflect the direction of the moon's 
motion through four right angles in 2 7* 32 2 days, the 
moon moving with a velocity which we may represent 
by ^^P^.* Now the sun at a distance from the earth 
equal to about 91,500,000 miles, has power to deflect 
the direction of her motion through four right angles 
in 365'256 days, the earth moving with a velocity 
which we may represent by ^^~^' ^ow, first, since 
gravity varies inversely as the square of the distance, 
the sun would require (if other things were equal) to 
have an attractive power exceeding the earth's in the 
ratio ( w^vv^)^ to produce the same eSect on her that she 

^ 23o,o00 ' -L 

produces on the moon ; and secondly, since the deflec- 
tion of a body's *line of motion is a work which will be 

* W need not consider the velocity in miles per hour, or the 
like ; because, throughout the paragraph, relative and not absolute 
velocities are in question. Hence we can represent the moon's 
velocity by the radius of her orbit divided by her period, provided 
we represent the earth's velocity round the sun in like manner. 


done at a rate proportional to the force wliicli operates, 
the sun's power (if other things were equal) should be 
less than the earth's in the ratio 3^^, to accomplish 
in one year what the earth accomplishes in a month ; 
and, lastly, since the faster a body moves the greater 
is the force necessary to deflect its course through a 
given angle in a given time, it is obvious that the 
sun's attractive power should exceed the earth's in 
the proportion of ''^iS? to 1S?.~ tliat is, in the ratio 

J- J- 000 zoo 27 o*2'2 

^'ir^^SSs *^ produce a given change of direction in 
the case of the quickly -moving earth in the same time 
that^ the earth produces such a change in the case of 
the less-swiftly-moving moon. Now, we have only to 
combine these three proportions,"^ which take into 
account every circumstance in which the sun's action 
on the earth differs from the earth's action on the 
moon, in order to deduce the relation between the 
sun's attractive energy and the earth's, at equal 
distances from the centre of either. This gives 
the proportion (^^")^x i^^Y, which reduces to 
314,798, in which proportion the sun's attractive 
energy exceeds the earth's. We may take 315,000 
as representing this proportion in round numbers, 
with an accuracy at least equal to that with which the 
sun's distance has been determined. 

Now in order to see whether the sun or the earth 
has the greater influence on the moon, we have only 
to compare the masses of the first-named two orbs 

* The whole process corresponds exactly to an ordinary problem 
in double (or rather multiple) rule of three. 


and tlie influence of tlieir respective distances from 
the moon. We thus have^^irs^, the proportion 315^000 
to 1^ in which the sun^s attraction exceeds the eartVs. 
at equal distances ; and secondly, the proportion 
(238,800)^ to (91,500,000)2 in which the attraction 
due to the sun's distance falls short of that due to 
the earth^s. Thus we have this relation, the sun^s 
actual influence on the moon bears to the earth^s 
the proportion which 314,500 x (238,800) ^ bears to 
(91,500,000) =*, or approximately a proportion of 15 
to 7.* Thus the sun^s influence on the moon is more 
than twice as great as the earth^s. 

It may be asked, then, how it is that the moon does 
not leave the earth^s company to obey the sun's 
superior influence ? In particular it might seem that 
when the moon is between the earth and the sun (or 
as placed at the time of a total eclipse), our satellite 
being then drawn more than twice as forcibly from 
the earth towards the sun as she is drawn towards the 
earth from the sun, ought incontinently to pass away 
sunwards and leave the earth moonless. 

The answer to this enigma is, simply, that the sun 
attracts the earth as well as the moon, and with almost 
the same degree of force, his pull on the earth some- 
times slightly exceeding, at others slightly falling 
short, of his pull on the moon, according as the dis- 
tance of the moon or earth from him is greater at the 

* The actual proportion, is 21421 correct to the fourth decimal 
place. The proportion 15 to 7 is equal to 2-1429, which for ordi- 
nary purposes is sufficiently near. 


moment. Thus the earth_, in order to prevent the 
escape of her satellite, has not to overcome the sun's 
pull upon the moon, but only the excess of that pull 
over the pull he exerts upon the earth herself. This 
excess, as will presently appear, is always far less than 
the earth's own influence on the moon. 

But it may be noticed, that in considering the moon's 
course round the sun we recognize the inferiority of 
the earth's influence in a very evident manner. The 
moon seems well under the earth's control when we con- 
sider only the nature of the lunar orbit round the earth ; 
but if for a moment we forget the fact that the moon 
is circling round the earth, and consider only the fact 
that the moon travels as a planet round the sun, with 
perturbations produced by the attractions of another 
planet, our own earth, 'we can readily test the extent 
of these perturbations. Now let the circle M M' (fig. 16, 
Plate y.) represent the moon's path round the sun S, 
and let us suppose that at the moon is between the 
earth and sun, and again similarly placed at 1, 2, 3 ... . 
11, and 12, being therefore on the side away from the 
sun at the intermediate stations marked with a small 
line outside the circle M M' ; then the moon's orbital 
course is a serpentine or waved curve, having its 
minima of distance from the sun at 0, 1, 2, 3 .... 11, 12, 
and its maxima of distance at the intermediate points. 
But on the scale of fig. 16, the whole of this serpentine 
curve would lie within the breadth of the fine circular 
line MM'. Thus it will readily be understood that 
the curvature of the moon's path remains throughout 


concave towards S, even when^ as at the points Oj !_, 2, 3, 
&c., the convexity of the orbital path round the earth 
is turned directly towards the sun. In other wordsj 
as the moon travels in her orbit round the sun her 
course is continually being deflected inwards from the 
tangent line, or always towards the sun. It is to be 
noticed, however, that the earth's perturbing influence 
is an important element in determining the moon's real 
orbit. For when the earth and sun are on the same 
side of the moon, or at the time of full moon, the pull 
on the moon is the sum of the pulls of the earth and 
sun, or exceeds the sun's pull alone in the ratio 22 
to 15 ; and on the other hand, when the earth and sun 
are on opposite sides of the moon, or at the time of 
new moon, the pull on the moon is the difi'erence of 
the pulls of the sun and earth, or is less than the sun's 
pull alone in the proportion of 8 to 15. Thus at the 
time of full moon the moon is acted on by a force 
which exceeds that acting on her at the time of new 
moon in the ratio of 22 to 8 or 11 to 4. And though 
at the time of full moon the moon's actual velocity 
(that is, her velocity in her orbit round the sun) is at 
a maximum, being then the sum of her mean orbital 
velocity round the sun and of her velocity round the 
earth j yet this by no means counterbalances the 
effects of the greatly increased pull on the moon :* so 

* The earth's velocity in her orbit being about 65,000 miles per 
hour, and the moon's about 2,000 miles per hour, the extreme . 
variation of the moon's motion in her orbit round the sun lies 
between the values 67,000 and 63,000 miles (roughly), or about four 


that the curvature of her path when she is ^^fulP' 
greatly exceeds the curvature at the time of new 

It was necessary to say so much about the moon's 
path round the sun, and the sun's real influence upon 
our satellite^ because a great deal of confusion very 
commonly prevails in the student's mind on this sub- 
ject. He is exceedingly apt^ when his attention is 
chiefly (and in the first instance) directed to the sun's 
perturbing influence^ to suppose that our earth plays 
the chief part in ruling the motions of the moon^ 
whereas the sun's influence is in reality paramount 
at all times. 

In considering the moon's motion around the earth, 
however, we may leave oiit of consideration the com- 
mon influence of the sun upon both these orbs, and 
need consider only the difference of his influence upon 
the earth and moon, since this difference can alone 
affect the moon's motion around the earth. 

Now we are enabled to deal somewhat more readily 
with this case than with the general problem of three 
bodies, because the moon is always very close to the 
earth as compared with the distance of either from the 
sun. On this account lines drawn to the sun from the 
earth and moon enclose so small an angle that they 
may be regarded as appreciably parallel. Again, these 
lines are at all times so nearly equal, that in deter- 
mining the relative pull on the earth and moon we 

times, or in the ratio of 110 to 103. But the attractive force on 
the moon varies in the ratio of 110 to 40, as above shown. 


^(jf 10 . 1 1 I ushriUn/j theMooji^s /rwtwn nmnrl /lie /Sun. 


^LProchr r/e/} 


may employ a simple method available with quantities 
that are nearly equal. Thus_, suppose two bodies placed 
at distances represented by. 100 and 101 respectively 
from a certain centre of force, then the attractions in the 
two bodies are inversely proportional to the squares of 
lOOand lOl.or are in the ratio 10,201 to 10,000; but this 
ratio is appreciably the same as the ratio of 102 to 100. 
Therefore in this case, and in all such cases * where 
the distance from one body exceeds the distance from 
the other by a relatively minute quantity, we can 
obtain the relative forces by representing them as lines 
having a relative difference twice as great. 

Now let us apply this principle to the moon and 
earth. Suppose E (fig. 17, Plate Y.) to be the 
earth, M the moon, and that the lines B 5, M / 
(appreciably parallel) are directed towards the distant 
sun. We may suppose the globe S to represent the 
sun, and we may regard S s' and S s as the pro- 
longations of M / and E 6-, if we recognize the fact 
that the gap at ss, ss, would, on the scale of our 
figure be some ten yards across. Now suppose that 
the sun^s attraction on a unit of the moon's mass f is 

* The student should test this assertion by a few calculations. 
Thus he can take the numbers 45,681 and 45,682, and show that 
the ratio of the squares of these numbers is approximately repre- 
sented by the ratio 45,681 to 45,683 ; and therefore the inverse 
ratio of the squares by the ratio 45,683 to 45,681. We may equally 
well take 45,680 to 45,682 for the ratio of the squares, and 45,682 
to 45,680 for that of the inverse squares. 

t Throughout the explanation it must be carefully borne in 
mind that when the attraction on the moon or earth is spoken of, 


represented by the line joining S and M, then the 
line joining S and E will be too large to represent 
the sun^s attraction on a unit of the earth's mass, for 
E is farther away from S than M is (in the state of 
things represented by the figure), so that the attrac- 
tion on E is less than the attraction on M. If we draw 
M K square to E s, we have the distance of K from S 
appreciably equal to the distance of M from S. K E is 
then the excess of the distance of E from S over the 
distance of M from S. If the sun's attraction dimi- 
nished as the distance increased, that is, if it were 
simply as the inverse distance, we need only take off 
K L equal to this excess E K, in order to get the line 
from S to L representing the attraction of the sun on 
the earth atE. But as t^ie force is inverselyas the square 
of the distance, we must (from what was shown in the 
preceding paragraph) take KH equal to twice the 
excess E K, in order to have the distance from S to H 
representing the sun's attraction on the earth at E. 

what is really to be considered is the attraction on each unit of 
the mass of either body. The attraction of the sun on the whole 
mass of the earth is always far larger than his attraction on 
the whole mass of the moon : but this circumstance in no way 
concerns us in studying the lunar perturbations. For that excess 
of attraction which depends on the earth's greater mass is strictly 
compensated by the circumstance that the mass affected by it is 
corrrespondingly great. The case may be compared to that of two 
unequal masses let fall at the same moment from the same height 
above the earth. Here the earth's attraction on the greater mass 
is greater than her attraction on the less. Yet the greater mass falls 
at no greater rate ; because that greater attraction is employed to 
move a correspondingly greater mass. 


Now, let us make a separate figure to indicate tlie 
actual state of things in such a case as we have con- 
sidered. There is the sun at S (fig. 18, Plate Y.) 
pulling at the moon with a force which we have repre- 
sented by M S ; and he is pulling at the earth with a 
force which we represent on the same scale by S H. 
This last force, so far as the moon^s place v^^ith respect 
to the earth is concerned, is clearly a force tending to 
keep the moon and earth together. It may be repre- 
sented then, in this sense, by the line S H, or as a force 
tending to thrust the moon from the sun (almost as 
strongly and directly as the direct action on M tends 
to draw the moon towards the sun).* Thus the moon 
is virtually acted on by the two forces represented by 
M S and S H, and therefore, by^ the well-known pro- 
position called the triangle of forces, we have as the 
resultant perturbing action on the moon, a force re- 
presented by the line M H.f 

* Of course the sun's action on the earth does not really amount 
to a force thrusting or repelhng the moon from the sun. But in 
determining the sun's perturbing action on the moon, we have in 
effect to take the excess or defect of the sun's full action on the 
moon, as compared with his full action on the earth, so that the 
latter action necessarily comes to be viewed in a sense contrary to 
its real nature, precisely as in ordinary arithmetic a sum which is 
positive in itseK comes to be viewed as negative when it is to be 

t If the student is more familiar with the parallelogram of forces 
than with the same property under the form called the triangle of 
forces, he should draw a line from M parallel and equal to S H ; he 
will find that M H is the diagonal of a parallelogram having this 
line and M S as adjacent sides. 

80 THE moon's motions. 

Thus we have an exceedingly simple construction 
for determining the sun's perturbing action on the 
moon (as compared with his direct action) when she 
is in any given position. We have merely to draw 
M K square to the line joining E and S, to take K H 
equal to twice E K, and to join M H; then M H is the 
perturbing force, where the line joining M and S repre- 
sents the sun's direct action on the moon."'*" 

Let us now figure the various degrees of perturbing 
force exerted on the moon when she is in different 
parts of her orbit, neglecting for the present the in- 
clination of her path to the ecliptic ; in other words, 
regarding all such lines as M H (fig. 18) as lying in one 
plane. The ellipticity of the moon's orbit is also for the 
moment neglected.' In fig. 19, Plate YI., this has been 
done. To avoid confusion, the differetit points where 
the action of the perturbing force is indicated have 
not been all lettered. Nor has the construction for 
obtaining the lines indicating the perturbing force 
been indicated in any instance. The student will, 
however, have no difficulty in interpreting the figure. 
Ml M2 M-jM^is the mood's orbit around the earth at E. 
The sun is supposed to lie on the right in the pro- 
longation of E A. At Ml the perturbing force is 
outwards towards the sun, and is represented in 
magnitude and direction by the line M^ A, which is 

* Practically M H may be taken to represent the sun's perturb- 
ing action on the moon when the line joining E and S represents 
the sun's direct action on the earth ; for the proportion of M S to 
either E S or H S, is very nei.rly unity under all circumstances. 



Fia 19 . Sfiewi?i^ me total Jorees joe?^iiirbina the J^oon 

Fta ?2. 'Tlw nadial parts of the same pertiirhinojorces. 

ria23. The Tametitml jjarts of the same joe?^tur/jimJo?Tes. 

On.thecfml(>ofI^sJ9,22,(il2SJJieJ'iaiihhatlrac(i^ o/( /he 
Moo7t wouU te repre^mZed try t)OHr?is A A ' , the i Sa m tr// /2SH}iiesAA 
KJ.Froaor rtet! ~ " 


equal to twice E Mj. As tlie moon passes from M^ to 
Oi the perturbing force gradually becomes more and 
more inclined to the line EA, but continues to act 
outwards with respect to the orbit M^ Mg M3 M4. At 
Oi,* however, the perturbing force is for the moment 
tangential to the orbit, and afterthe moon has passed Oi, 
the force acts inwards. This continues until the moon 
has passed to O2, a point corresponding in position to 0^ 
but on the left of Mg E. At Mg it is clear that the 
force is represented by the line MgE, and is simply 
radial. Also, in actual amount the perturbing force is 
less at Mg than at any other point in the semicircle 
Ml M2 Mg. After passing Og the force is again exerted 
outwards, becoming wholly outwards at M3, when it is 
represented by the line M3 A' equal to M^ A, or to the 
diameter of the circle Mi M2 Mg M4. Passing from 
Mg to M4, and thence to Mi, the moon is subjected to 

* Oj is determined by the circumstance, that when Oi K is 
drawn square to EA (the student should pencil in the lines 
and letters here mentioned), and KL taken equal to twice 
E K, Oi L is a tangent to the circle Mi M2 M3 M4. Since the 
square on line E Oi is equal to the rectangle under E K, E L, or 
to three times the square of E K, we obviously have the cosine 

of the angle Oi E K equal to -7=, whence Oi Mi is an arc of 54 44' ; 

and O2 Mg, M3 O3, and O4 M^ are also arcs of 54 44'. In Herschel's 
Outliyies of Astronomy these arcs are given as 64 14', and the 
figure to art. 676 is correspondingly proportioned. But 54 44' is 
the correct value. Indeed it will be obvious in a moment that an 
arc of 60 would give a perturbing force lying within the tangent, 
since the tangent at the extremity of an arc of 60 clearly cuts the 
line E A at a distance from E four times as great as the distance 
of the foot of a perpendicular let fall from the same extremity. 



corresponding perturbing forces, varying in the reverse 
way. At O3 the force is wholly tangential, at M^ it is 
wholly radial, and represented by the line M4 E. At 
O4 it is again wholly tangential ; and, lastly, at M^, the 
force is again wholly radial as at first, and represented 
by the line M^ A. 

It will be very obvious that, on the whole, the per- 
turbing action tends to diminish the earth's influence 
on the moon, since the forces acting outwards are 
greater in amount, and act over larger arcs than those 
acting inwards. We see that M^ A and Mg A', the 
maximum outward forces, are twice as great as Mg E 
and M4 E, the maximum inward forces ; while the arcs 
O4 Oi and O2 O3 each contain 109 28', or in all 
nearly 219 out of 360, that is, more than three- 
fifths of the complete circumference. Hence we can 
infer that there is a consider ib\) balance of force 
exerted outwards. 

But it will be well to picture the radial forces 

Let M H, fig. 20, represent the disturbing force 
on the moon at M, and draw E M B radially and M C 
tangentially. Then complete the rectangle B C by 
drawing the perpendiculars H B and H C. By the 
well-known rule for the resolution of forces, the force 
M H is equivalent to the two forces represented by 
M B and M C, one radial, the other tangential. Simi- 
larly, if we had commenced by considering the force 
M' H', fig. 21, exerted on the moon at M', we should 
have found by a similar construction that the radial 


and tangential forces at M' are represented by the 
lines M^ B' and M' C ; and so on_, for all positions. 

Leaving the tangential forces for subsequent con- 
sideration_, let us suppose the above construction 
extended so as to give the radial forces exerted at 
points all around the moon's orbit. We should then 
have the result pictured in fig. 22, the radial forces 
all exerted outwards from O4 to O^, and from Og to O3, 
while they are all exerted inwards from Oj to O2, and 
from O3 to O4. We see that the former forces largely 
exceed the latter. 

The first great result, then, from the consideration 
of the moon's perturbing action, is this, it tends 
to draw the moon on the whole outwards from the 
earth, reducing the earth's influence to a certain extent. 

We can compare the actual amount of the radial 
force (or of the perturbing force geror-i'ly) with the 
amount of the earth's attraction ; and it is important 
that we should do so in order that we may judge how 
the forces acting on the moon are related as respects 

For it will be remembered that the construction for 
obtaining fip'. 19 is based on the supposition that the 
line from E to the sun represents the sun's direct 
attraction on the earth or moon. Now, the line from 
the earth to the sun is about 91,500,000 miles long; 
while the line M^ A is equal to the diameter of the 
earth's orbit, or to 238,800 miles. So that the sun's 
maximum perturbing action on the moon is less than 
his direct action, in the proportion of 2,388 to 915,000, 
G 2 


or is about one-383rd part of the latter. But tlie 
earth's direct action on the moon is^ as we have seen, 
equivalent to about 7-15ths of the sun's. Hence the 
sun's maximum perturbing influence is less than the 
earth's mean attraction on the moon, in the proportion 
of 15 to 7 X 383^ or is about one-179th part of the 
latter. Thus the force pulling the moon at Mj towards 
the sun, would be represented by a line 383 times as 
long as Mj A, while the force pulHng the moon to- 
wards E would be represented by a line 179 times as 
long as Ml A. The relations for the perturbing forces 
exerted on the moon in other positions, as well for the 
whole forces as for their radial and tangential portions, 
are indicated by the proportions of the lines in figs. 
19, 22, and 23. When the perturbing force has its least 
value, or when the moon is at Mg or M4, this force, 
now wholly radial, is about one- 776th of the sun's 
direct action, and about one-358th of the earth's. 

But now we have to consider the circumstance that 
the earth's path around the sun is eccentric, and that 
thus the sun's perturbing influence on the moon 
necessarily varies in amount. It will be obvious that 
the perturbing forces must all be greater when the 
earth and her satellite are nearer to the sun. Let ua 
inquire in what degree they will increase. 

This question is readily answered. Fig. 19 indi- 
cates the magnitude of the perturbing forces when 
the line from the sun to E indicates the sun's direct 
action. Now to simplify matters let us take an illus- 
trative case, in order to determine the law according 

THE moon's motions. 85 

to whicli the magnitude of the perturbing forces are 
affected. We have hitherto supposed the earth at her 
mean distance from the sun, or about 91,500,000 
miles from him. Let us now take the case when she 
is in perihehon, or about 90,000,000 miles from him. 
The moon^s distance, 238,800, is contained a smaller 
number of times in the smaller distance, in the pro- 
portion of 900 to 915 j in other -words, the perturbing 
force represented by Mj A is a larger aliquot part of 
the sun's direct influence, in the proportion of 915 to 
900. But the sun's direct influence is itself increased 
by the approach of the earth and her satellite, in the 
proportion of the squares of these numbers ; or as 
(915)^ to (900)^. Hence the actual amount of the 
perturbing force is increased in the proportion of 
the cubes of these numbers, or as (915)^ to (900)^. 
Similarly when the earth is in aphelion, or 93,000,000 
miles from the sun, the sun's perturbing influence is 
less than when the earth is at her mean distance, in 
the proportion of (900)^ to (930)^ 

There is, however, a simpler method, sufficiently 
accurate for our purposes, of indicating these rela- 
tions. When we cube two numbers which are nearly 
equal, we triple the proportional difference (approxi- 
mately). Thus if we cube 100 and 101 (whose dif- 
ference is 1-1 00th of the former) we obtain the 
numbers 1,000,000 and 1,030,301, which are to each 
other very nearly as 100 to 103; so that their dif- 
ference is about 3-lOOths of the former. Now the 
earth's greatest, mean, and least distances from the 


sun are approximately as the numbers 62, 61, and 60; 
and therefore the perturbing influences on the moon 
when the earth is in aphelion, at mean distance, and in 
perihelion, are respectively as the numbers 64, 61, and 
58 (obtained by leaving the middle number 61 un- 
altered, and making the first and last difl'er three 
times as much as before from the middle number). 

There is, then, an appreciable difference between 
the perturbing forces exerted by the sun when the 
earth is in perihelion, or at about the beginning of 
January, and when the earth is in aphelion, or at 
about the beginning of July. The earth^s power over 
the moon is more considerably diminished in the 
former case than in the latter. Now the partial 
release of the moon from the earth^s influence results 
in a slight increase of her mean distance and a 
lengthening of the moon^s period of revolution (we 
refer of course to her sidereal revolution) around the 
earth. This will be evident when we consider that 
the earth^s attraction is always tending, though the 
tendency may not actually operate, to reduce the 
moon^s distance; so that any cause diminishing the 
total force towards the earth must enable the moon to 
resist this tendency more eff'ectually than she other- 
wise would. In winter, then, when the earth is near 
perihelion, the moon's mean distance and her period of 
revolution are somewhat in excess of the average ; for 
the sun's releasing effect is then at a maximum. In 
summer, on the contrary, the earth being near aphelion, 
the moon's mean distance and her period of revolution 

THE moon's motions. 87 

are reduced slightly below their mean values ; for the 
sun's releasing effect is then at a minimum. Thus the 
moon lags somewhat during the winter months, and 
regains her place by slightly hastening during the 
summer months. She is farthest behind her mean 
place, so far as this circumstance is concerned, in 
spring and autumn (at those epochs when she is at 
her mean distance), for it is at these times that the 
loss begins to change into gain, or vice versa. The 
greatest possible amount of lagging accruing in spring 
is such that the moon is behind her mean place by 
about a third of her own diameter. In autumn she 
gets in advance of her mean place by about the same 

This peculiarity of the moon's motion is called the 
annual equation, and was discovered by Tycho Brahe. 

Associated with this variation is another of much 
greater delicacy, and having a period of much greater 
length. We have seen that the eccentricity of the 
earth's orbit affects the amount of the sun's perturbing 
influence, insomuch that this influence is sometimes 
greater and sometimes less than when the earth is at 
her mean distance. It might appear that as there is 
thus an excess at one season and a defect at another, 
the general result for the year would be the same as 
though the earth travelled in a circular orbit at her 
present mean distance from the sun. This, however, is 
not the case. If we consider that, supposing the earth 
to revolve always at her mean distance she would 
describe a circle having a diameter as great as the 


major axis of her actual orbitj we see that the elliptical 
area of her real path is less than that of the supposed 
circular orbit. Hence, on the whole, she is nearer to 
the sun than if she described a circular orbit in a year 
instead of her elliptical path. It is true that she 
moves more slowly when in aphelion, and thus her 
virtual yearly distance (so to speak) from the sun is 
increased; but this does not compensate for the actual 
reduction of her orbit-area due to the eccentricity of 
her orbit.* Hence the sun^s perturbing influence on 
the moon is somewhat greater, owing to the ellipticity 
of the earth^s orbit. Now this ellipticity is subject 
to slow variation, due to the influences of planetary 
attraction. At present it is slowly diminishing. The 
earth^s orbit is slowly becoming more and more nearly 
circular, without, however, any change (or any cor- 
responding change) in the period of revolution. Thus 
the area swept out by the earth each year is slowly 
increasing, and the total of the sun^s perturbing in- 
fluence on the moon in each year is slowly diminishing. 
The moon then is somewhat less retarded year after 
year; so that in efi'ect she travels somewhat more 
quickly year after year. This change is called the 
secular acceleration of the moon's mean motion , or 
rather an acceleration which is partially accounted for 

* The reasoning by which this may be demonstrated corre- 
sponds precisely with that in pp. 166, 167 of my treatise on 
Saturn, where I show that a planet receives more heat during 
a complete revolution in an elliptical orbit, than it would receive 
in revolving round a circular orbit in the same period. 


by the above reasoning has received this name. As a 
matter of fact, the moon's mean motion is subject to 
an acceleration nearly twice as great as the change in 
the elhpticity of the terrestrial orbit will account for; 
and astronomers have been led to suspect that a por- 
tion of this acceleration may be only apparent and 
due to a real retardation of the earth's rotation, that 
is, a slight increase in the sidereal day, the unit by 
which we measure astronomical time. With this cir- 
cumstance, however, we are not at present concerned, 
save in so far as it relates to the history of that 
interesting cause of acceleration which has been de- 
scribed above. Halley had been led to suspect that 
the moon had advanced somewhat farther in her 
orbit than was consistent with the accounts of certain 
ancient eclipses.* Further inquiries confirmed the 

* I quote here some remarks on Halley's researches by Mr. J. M. 
Wilson, of Rugby, from a valuable paper contributed to The Eagle, 
a magazine supported by members of St. John's College, Cambridge 
(No. xxvi. vol. v.). " Halley," he says, " seems to have been the 
first who considered this question. With astonishing clearness he 
seized the conditions of the question, saw that the knowledge of 
the elements, on which the solution was to be founded, was as yet 
incomplete, and saw also the probability that when the accurate 
knowledge was obtained, it would appear that there was a pecu- 
liarity in the moon's motion entirely unforeseen by others, that it 
was now moving faster and performing its revolution in a shorter 
time than it did in past time. If the longitudes of Bagdad, 
Antioch, and other places, were accurately known, ' I could then,' 
he says, ' pronounce in what proportion the moon's motion does 
accelerate ; which that it does, I think I can demonstrate, and 
shall (God willing) one day make it appear to the public' Newton 
adds to his second edition of the Principia the words, ' Halleius 


suspicion. The moon's advance was slight^ it is true, 
but to the astronomer it was as real as though it had 
taken place under his very eyes. The theory of gra- 
vitation seemed to give no account of this acceleration 
of the moon^s motion. At length, however, Laplace 
was led to turn his attention to the variation of the 
earth^s eccentricity as a probable cause of the pecu- 
liarity. His calculation of the effects due to this vari- 
ation accorded very closely with the observed amount 
of the acceleration. Yet, although this agreement 
might have appeared convincing, and although a por- 
tion of the acceleration is undoubtedly due to the 
cause in question, the inquiries of Professor Adams 
(confirmed by the researches of Delaunay and others, 
and now universally admitted) show that in reality 
only half the observed acceleration can be explained 
by the change in the earth^s orbital eccentricity. 

But it is to be noted that the variation itself is 
exceedingly small, as is also the discrepancy between 
observation and theory. We have seen that the 
annual equation causes the moon to be displaced by 
about one-third of its diameter in opposite directions 
in spring and autumn, the actual range of this oscil- 
latory variation being therefore equal to about two- 

noster motum medium Lunse, cum motu diurno terrae coUatum 
paulatim accelerari primus omnium quod sciam deprehendit.' " 

I have given an account of the subject in the Quarterly 
Journal of Science for October, 1866, in an essay entitled " Prof. 
Adams's Kecent Discoveries," and a more popular account appears 
in my Light Science for Leisure Hours, in a paper called "Our Chief 
Timepiece losing Time." 


thirds of the moon^s diameter. But the theoretical 
secular acceleration, though its effects are accumu- 
lative, and in geometrical progression, yet in a cen- 
tury would only cause the moon to be in advance of 
the place which she would have had, if the accelera- 
tion had not operated during the century, by one- 
300th part of her diameter ; and the actual secular 
acceleration only causes the moon to gain about twice 
this distance, or about one-150th part of her diameter, 
in a century.* 

We have next to consider one of the most im- 
portant perturbations to which the moon is subjected so 
far as the rate of her motion in her orbit is concerned. 

We have hitherto considered chiefly the radial part 
of the perturbing force. We must now discuss the 
variations in the tangential force. We have already 
seen how this force can be separated from the radial 
force (see p. 82). Let us suppose the method applied 
to give a figure of the tangential forces correspond- 
ing to that already given (fig. 22) for the radial forces. 
To do this, we have to draw a number of lines obtained 
as M C and M' k! were obtained in figs. 20, 21. When 
this is done (and the reader is recommended not to 
be satisfied until he has effected the construction for 
himself independently), the force-lines are found to 
arrange themselves as shown in fig. 23. It will be 

* In two hundred years the gain is four times as great, in three 
hundred years nine times as much, and so on. For the above illus- 
tration I am indebted to Mr. Wilson's paper mentioned in the 
preceding note. 


seen tliat each loop springs from one of the points M^, 
M2, Mg, M4 (where the tangential force vanishes_, and 
the radial forces have their unequal maxima), and 
bends round so as to end at another of those four 
points ; and we see that at the four points m^, m^,, m^, 
m,^ (midway between the former, and not far from 
those where the radial force vanishes) the tangential 
force has its maximum value.* 

Now as the moon is passing over the arc M^ Mg, the 
tangential force, acting in the direction shown by the 
curves, is retardative, and most effectually so when the 
moon is in the middle of this arc, or at the point Wj. 
As the moon passes from Mg to Mg, the tangential 
force is accelerative, and most effectually so when the 
moon is at the middle of the arc M2 Mg, or at the point 
mg. As the moon passes over the arc M3M4, the 
tangential force is again retardative ; and it is again 
accelerative as the moon traverses the arc M4 M^, 
attaining its greatest value when the moon is at the 
middle of these respective arcs, or at mg and m^. 
Since, then, retardation ceases to act when th^ moon 
is at Mg, the moon is moving there with minimum 
velocity, so far as this cause of disturbance is con- 
cerned. In like manner the moon is moving with 
maximum velocity at Mg, with minimum velocity at 

* The tangential force attains its maximum midway between 
the points Mi, M2, M3, M4, and not, as is sometimes stated, at the 
points where the radial force vanishes. It will be obvious from 
fig. 20 that if we call the angle H E M, 0, we have H B=3 cos 
sin =4 sin 29 ; and this expression has its greatest value when 


M4, and lastly with maximum velocity again at M^. 
It will be clear, then, that near the points m^, m.^, Wg, 
and m^, the moon moves with mean velocity ; the arcs 
m^ m^ and m^, m^ are traversed with a velocity exceeding 
the mean ; and the arcs m^ m^ and m^ m^ with a velocity 
falling short of the mean. Thus at or near m^ the moon 
ceases to gain, and therefore the amount by which she 
is in advance of her mean place has attained its max- 
imum * when the moon is at or near m^. Similarly the 
amount by which the moon is behind her mean place has 
attained its maximum when the moon is at or near mg. 

* It is singular how frequently the very simple principles on 
which the attainment of a maximum, mean, or minimum value 
depend are misunderstood ; and how commonly the mistake is 
made of supposing that a maximum or minimum value is attained 
when the increasing or diminishing cause is most effective. It 
is precisely when an increasing cause is most effective that the rate 
of increase is greatest, and therefore the maximum value is then 
clearly not attained. And so of a minimum value ; there can 
clearly be no minimum while the decreasing cause is still effect- 
ive. It is when a cause neither tends to increase nor diminish, 
that is, when it has a mean value, that the maximum or minimum 
of effect is attained. Thus in spring the sun's daily elevation is 
increasing more rapidly than at any other time, and in autumn the 
daily elevation, is diminishing most rapidly ; but it is not at these 
seasons that the sun attains his maximum or minimum degree of 
elevation ; this happens in the summer and winter, when his daily 
elevation changes least. In an illustrative case such as this there can 
be no mistake ; yet very often, when less familiar instances are dealt 
with, the mistake to which I have referred is made. Thus in Mr. 
Lockyer's Elementary Lessons in Astronomy, we have the seasons 
when the equation of time is zero described as those when the 
real sun's motion is the same as the mean sun's ; the fact really being 
that it is precisely at these seasons that the real sun's motion attains 
either its maximum or minimum value. 


At Wg she is again in advance of her mean place by a 
maximum amount^ and at w^ she is again behind her 
mean place by a maximum amount. 

This inequality of the moon^s motion is called the Fan- 
ation. It is so marked that at the points corresponding 
to m^ and m^ the moon is in advance of her mean place 
by an amount equal to about her own diameter, while 
at mg and m^ she is by a similar amount behind her 
mean place. The range of the variation is thus equal 
to about twice the moon^s diameter. The period of 
the variation is on the average half a lunation, since 
in that time the moon passes from her greatest re- 
tardation (due to this cause) to her greatest advance, 
and so back to her greatest retardation. We owe to 
Tycho Brahe the discovery of this inequality in the 
moon's motion.* 

And now, precisely as we had, after considering the 
annual equation, to discuss an associated but much less 
considerable inequality, so there is an inequality asso- 
ciated with the variation, but much smaller in amount. 
It is, however, more interesting in many respects, 
precisely as the secular acceleration of the moon is a 
more interesting inequality than her annual equation. 

We have hitherto not taken into account the cir- 
cumstance that though the sun's distance euormously 

* It will be evident that the ancients, who trusted chiefly to 
eclipses to determine the laws of the moon's motion, were pre- 
cluded from recognizing the remarkable displacement due to the 
variation ; since eclipses necessarily occur when the moon is on the 
line passing through the earth and sun, or when the moon is at 
Ml or Ms, at which points the variation vanishes. 


exceeds the radius of the moon^s orbit, it is neverthe- 
less not so great but that there is an appreciable 
relative difiPerence between the moon's distance from 
the sun when in conjunction with him (or at the time 
of new moon), and when in opposition (or at the time 
of full moon). When the earth is at her mean dis- 
tance from the sun (or 91,600,000 miles from him), 
the moon's distance from him when she is new is 
91,738,800 miles, and when she is full it is only 
91,261,200 miles, so that these extreme distances 
are proportioned as the numbers 917,388 and 912,612, 
or, nearly enough for our purposes, they are as the 
numbers 201 and 200. Hence, by what has been 
already shown, the perturbing forces on the moon in 
these two positions are as the numbers 203 and 200. 
Thus the difference, though slight, is perceptible. 
Yet again, the points where lines drawn from the 
sun touch the moon's orbit are not quite coincident 
with the points Mg and M4 (fig. 19), but are slightly 
displaced from these positions towards M^. Here, 
agaiu, the amount of either displacement, though 
slight, is appreciable. It amounts, in fact, to an arc 
of about 8f minutes ; so that the points in question 
divide the moon's orbit into two unequal arcs, whereof 
one, the farthest from the sun, exceeds a semicircle 
by 17| minutes, the other falling short of a semicircle 
by the same amount, the larger thus exceeding the 
smaller by 35', or more than half a degree. 

It necessarily follows that the direct effect of the 
tangential force in increasing or diminishing the 


moon's mean motion^ is not equal in tlie two halves 
M4 Ml Mg and Mg Mg M4 (fig. 19). It is greater in the 
former semicircle, on the whole, than in the latter ; 
but the points where the tangential force vanishes 
lie outside the extremities of this latter semicircle. 
Thus the points where the variation attains its maxi- 
mum value lie on the sides of m^ and m4 towards M^, 
and on the sides of m^ and m^ away from M3; 
and the amount of the maxima at the two former 
stations is greater than the amount at the two latter. 
Moreover, when the earth is in perihelion these effects 
are greater, while when she is in aphelion they are 
less than when she is at her mean distance. The 
maximum inequality thus produced, a variation of 
the variation as it were, amounts to about two 
minutes, or about the sixteenth part of the moon\s 
apparent diameter. It is called the parallactic 
inequality^ because of its dependence on the sun's 
distance, which, as we know, is usually expressed 
by a reference to the solar parallax. And as the 
inequality depends on the sun's distance, so its 
observed amount obviously supplies a means of de- 
termining the sun's distance. It was, in fact, a 
determination of the sun's distance, deduced by 
Hansen from the observed amount of the moon's 
maximum parallactic inequality, which recently led 
astronomers to question a value of the distance, 
based on observations of Yenus in transit, which 
had been for many years adopted in our text-books 
and national ephemerides. 


Before passing frOm the consideration of the direct 
action of the tangential force, it is to be noticed 
that this force affects the secular acceleration of the 
moon. It had long been held that only the radial 
force can really be effective in long intervals of time, 
because the tangential force is self-compensating, 
if not in each lunation,* yet at least in the course of 
many successive lunations. But as a matter of fact, 
inasmuch as the eccentricity of the earth's orbit is 
undergoing a continual though very gradual diminu- 
tion, an element is introduced which renders this 
compensation incomplete, not merely in many suc- 
cessive lunations or in many successive years, but in 
many successive centuries. So long as the eccen- 
tricity of the earth's orbit continues to diminish, there 
can in fact be no tendency to exact compensation so 
far as this particular element is concerned. Now 
this circumstance had not escaped Laplace when 
he discussed the moon's secular acceleration; but 
he was led to believe that its effects would be wholly 
insignificant. Professor Adams, however, in re- 
examining the whole subject, was led to inquire 
how far this view of the matter is justified. The 
experience of past inquirers had shown that no cause 
of variation, and particularly no cause having effects 

* Of course, in a thorough analysis of the action of the tangen- 
tial force, it has to be remembered that, apart from the ellipticity 
of the moon's orbit, and the consequent inequality of the sun's per- 
turbing action in different quadrants as well as in diiferent halves, 
the orbit is undergoing a process of continual change, even under 
the action of the tangential and radial forces themselves. 



cumulative for many successive years, can safely be 
neglected. Professor Adams remarked, " In a great 
problem of approximation, such as that presented to 
us by tbe investigation of the moon's motion, ex- 
perience shows that nothing is more easy than to 
neglect, on account of their apparent insignificance, 
considerations which, ultimately prove to be of the 
greatest importance.''^ We shall see presently how 
Newton himself fell into an error precisely resembling 
that of Laplace, in fact so far identical in its nature 
that it was the tangential force that Newton, like 
Laplace, held to be self-compensatory, thougb the 
instances to wbicb this erroneous consideration was 
applied were altogether distinct in their nature.* 

* It is a somewhat curious circumstance, that while the correc- 
tion applied by Adams to Laplace's labours resulted in reducing 
the theoretical secular acceleration by one-half, so the correction 
applied by Clairaut to Newton's inquiry into the motion of the 
moon's perigee resulted in doubling the theoretical amount of that 
motion. (Clairaut had himself, in the first instance, obtained by 
analytical researches the same erroneous value which, Newton had 
obtained from geometrical considerations.) It is to be remarked 
also that Adams's labours set theory and observation at apparent 
discordance after they had been brought into agreement, while 
Olairaut's labours brought theory and observation, which had long 
seemed discordant, into perfect agreement. N othing, perhaps, could 
more thoroughly demonstrate Adams's mastery of the lunar theory 
than his maintaining bis views against the great reputation of 
Laplace, seemingly also against obser^'^ation, and actually against 
the concurrent opinion of nearly all the greatest continental mathe- 
maticians. How slowly his views made ground will be seen from 
this, that the paper from which I have quoted was read before the 
Royal Society in 1853, and that it was not until the year 1866 that 


M. ,^J^^i ^i9^ 

J'i^28. Jfhstratimf ike actio rv of Mr?nat distwHTb^Jorces. 
Kl/IJ^roclo7^ Ml ^ 


Thus far we Lave considered those general effects 
only whicH could, be adequately discussed without any 
reference to the ellipticity and inclination of the lunar 
orbit. Now we have to examine the effects of the 
perturbing forces on the figure and position of the 
moon^s orbit, as well with respect to its ellipticity 
as to its inclination. 

In the first place it will be well to prevent a mis- 
apprehension which is very commonly entertained by 
'those who approach this subject for the first time. 
Looking at fig. 24, Plate YII.,and noticing the nature of 
the forces which are exerted upon the moon at different 
parts of her orbit, it seems natural to infer that since 
the moon is drawn outwards when at or near M^ and 
Mg, while she is drawn inwards when at or near Mg 
and M4, her orbit must necessarily be lengthened 
along the line M^Mg and narrowed alone the line 
Mg M4. In reality, the contrary happens. The forces 
exerted on the moon tend to diminish the curvature 
of the orbit at M^ and Mg, and to increase the curvature 
at Mg and M4. This is easily shown. We have seen 
that at Ml there is a radial perturbing force actiiig 
outwards, or resisting the earth's attraction on the 
moon. Hence if we suppose the moon to arrive at 
Ml as if moving on our hypothetical circular orbit, 
then instead of moving onwards to 0^ on this circle, 

the matter was so definitely settled in his favour, that the gold 
medal of the Astronomical Society was awarded to him. So late 
as 1861, we find his great rival Leverrier saying that "very cer- 
tainly the truth lay with Adams's opponents." 
H 2 


she would move on a patli sucli as Mj ]p^, being less 
strongly drawn towards the earth. Or if we suppose 
that she had arrived at O4 (the place where the radial 
force vanishes) on the circular orbit,, she would follow 
such a course as O4 c[^, on a curve less strongly curved 
than the circle O4 M^ Oi. Again at Mg the radial 
action reinforces the earth's attraction. Hence if the 
moon had arrived at Mg on the hypothetical circular 
path, then under the increased action due to the 
radial force, she would follow a path such as M2_p2i or 
such a path as 0^ q.2, if she had arrived at 0^ on the 
circular orbit : and it is obvious that both M2 ^2 ^^^ 
Oi ^2 ^1*6 more strongly curved than the circle O1M2O2. 
So that, partially freed from control over the arcs 
O4O1 and O2O3, the moon there tends to follow an arc 
of less curvature, a more flattened arc, so to speak, 
than in traversing the arcs 0^ O2 and O3 O4, where she 
is subjected to an increased radial force. 

Now let us inquire into the various circumstances 
which affect the position of the moon's perigee. 

We know that the radial force acts sometimes 
inwards and sometimes outwards, and that the tan- 
gential force is sometimes accelerative and sometimes 
retardative. Before proceeding to the actual relations 
of the lunar orbit, it will be well to consider the effect 
of radial and tangential forces thus acting. We 
ought, indeed, now that the. ellipticity of the moon's 
orbit is considered, to distinguish between a radial 
force and a force acting square to the tangential force; 
bu*5 the moon's orbit is not so eccentric as to render 


the distinction important in an inquiry such as the 
present. For the sake of brevity, and because in 
effect a complete inquiry would require a volume 
instead of a section of a chapter_, I take only the 
action on the moon when in perigee and apogee. 

Let us suppose that when the moon is at her perigee 
M, fig. 26, she is exposed to a radial perturbing force 
acting towards E, and let M K M' be the path which 
she would follow but for this perturbing force. Now 
if the attraction of the earth were suddenly but per- 
manently increased when the moon was at M, the 
path subsequently pursued by our satellite would be 
(so far as this cause is concerned) an ellipse, such as 
M L A, still having its perigee and apogee on the line 
M M^ But if, after the increased radial force had 
acted for awhile, say till the moon had reached L, 
it ceased to act, the moon^s orbit would as it were 
recover from its temporary contraction. The motion 
at L would be nearly the same as the motion at a 
point K on the undisturbed orbit (K being as far from. 
E as L is, and the tangent K T making appreciably 
the same angle with E K that the tangent L T makes 
with E L) . The velocity at L will also be appreciably 
the same as the velocity at K, for the arcs M L and 
M K are small, and the main effect of the radial dis- 
turbing force during the short time of its action, is 
that which it has had in drawing the moon inwards. 
Thus the line E K in the original orbit may be re- 
garded as having advanced to the position E L. The 
moon passes on from L under the same conditions as 

102 THE moon's motions. 

those under wliich slie would have passed from K if 
undisturbed. The orbit^ therefore, which the moon 
would traverse if thenceforth undisturbed is identical 
in shape with the orbit M K M'_, but differs in being 
shifted forwards so that E K has taken the position 
EL. EM then has taken the position E m in other- 
words, the perigee has advanced to the position m, 
and m E tyi is the new position of the major axis. 

An increase of the radial force, then, acting when 
the moon is in or near perigee causes the perigee to 
advance. And by like reasoning it may be shown 
that a diminution of the radial force acting near 
perigee, causes the perigee to regress.* 

Next let us take the case where the disturbing 
radial force acts on the moon when she is in or near 
apogee. Here, supposing that M' is the apogee of the 
lunar orbit and M' K M the undisturbed path, we have 
for the path which would result from a permanent in- 
crease of radial force, such an orbit as M' LA. But 
the disturbing increase ceasing when the moon is at 
L, we have the same conditions at L as at a point K 
on the original orbit (as far from E as L is, and 
having the tangent K T inclined at appreciably the 
same angle to E K as L T' to E L). Thus the position 
of the moon at L corresponds to a more advanced 

* We can regard M m L as the original orbit in this case, and 
M K as the orbit traversed under the reduced radial action. Then 
E K corresponds to E L, a more advanced position in the former 
orbit ; in other words, in this case each corresponding radial line 
in the new orbit (and therefore the line to the perigee) is farther 
back than in the old orbit. 


position at K in the original orbit. All other lines in 
the original orbit are similarly thrown backwards or 
caused to regredo. Thus the apogee is thrown to m 
and the perigee to m, and the new orbit has its major 
axis in the position m E m' . 

An increase of the radial force^ then, acting when 
the moon is in or near apogee, causes the perigee to 
regress. And by like reasoning it may be shown that 
a diminution of the radial force acting near apogee 
'causes the perigee to advance.* 

But a consideration of the reasoning in the cases 
just considered will show us how we may infer the 
effects of a change in the radial force when the moon 
is in other parts of her orbit. We shall, however, in 
what follows speak of the normal f force, because in 
fig. 28, and in figs. 29, 30, 31, &c., it is convenient to 
have an elliptic orbit of such a figure that there is a 
considerable difference between the direction of the 
radial line and of the normal (or perpendicular to the 
tangent). We remind the student, however, that in 
the actual case of the moon^s motions the radial and 
normal lines are always very nearly coincident. Now 
supposing the moon to be at P, fig. 28, when an 

* We may regard M' L as the original orbit in this case, and 
M' K as the disturbed orbit ; then E K in the latter corresponds 
to a less advanced line, E L, in the former, and thus every cor- 
responding line in the new orbit takes up an advanced position. 

t By the normal force is here understood the force acting per- 
pendicularly to the tangent. The actual normal force, of course, acts 
always inwards ; it is only the perturbation which acts sometimes 
outwards, diminishing the normal force, or inwards, increasing it. 


increase of the radial force is experienced, and to be 
travelling in the path P A' A, she would travel on a 
course touching her former path at the point P, but 
forming an ellipse smaller than P A A', if the radial 
force were permanently increased, and a path still touch- 
ing the former path in P, but larger than the ellipse 
PAA'_, if the radial force were permanently dimin- 
ished. But as the increase or diminution of the ra- 
dial, and therefore of the normal force, acts but for a 
short time, we have to consider that when the moon 
has traversed some small distance from P on the new 
path, the normal force is restored to its original value. 
In reality of course the normal force passes above and 
below its mean value with a continuous process of 
change, not starting suddenly from its mean to its 
maximum or minimum value : and in any thorough 
investigation intended to determine the quantitative 
effects of such changes, this circumstance must be 
taken into account. But in an inquiry such as the 
one we are upon, it is sufficient to consider the effects 
of an increase or diminution continuing to act during 
some short but definite time. Now it will be obvious 
that if exposed to an increased normal force, the 
moon, after travelling a short distance from P, would be 
moving on a course making a larger acute angle with 
P T than the course she would have had at the same 
epoch if undisturbed; whereas, if the normal force 
were diminished at P, the moon would be travelling 
on a course making a smaller acute angle with P T 
than the course she would have had at the same epoch 

THE moon's motions. 105 

if undisturbed. The alteration of tlie direction does 
not take place at P, or at any definite point on the 
moon's course ; but is the sum of the effects resulting 
either from increased or diminished radial action as 
the moon moves onwards from P. But, this remem- 
bered, we shall not err greatly if, to simplify the 
illustration, we suppose the alteration of direction to 
take place, per saltum, at the point P, even though P 
is the precise point where a permanent increase or 
diminution of the radial force would leave the tangency 
absolutely unaltered. A temporary change in the 
normal force does actually produce an alteration in 
the position of the orbit, which, if no further changes 
took place after the moon had travelled some distance 
from P, would affect the tangency of the orbit close 
by P.* The angle corresponding to S P T would be 
diminished if the normal force were increased ; while 
that angle would be increased if the normal force 
were diminished ; and the period of the orbit (or the 
major axis) would be unaffected, because the normal 
force would resume its original value, while the main 
effect produced on the moon's motion would be merely 
a change of direction. So that since the major axis 
of the orbit is equal to the sum of the lines S P, P H 
(H being the other focus), the new orbit would have 
its focus as far from P as H is, and therefore on the 
circular arc h H li about P as centre. And obviously, 
if the tangent P T took up the position P t, owing to 

" The moon's new orbit would not pass through P in that case. 


a diminislied normal force_, the new position of tlie 
other focus would be as at li, and S h would be the 
direction of the new axis ; while, an increased normal 
force causing the tangent at P to assume the position 
P if , would bring the other focus to such a position as 
at lb , S li being the new direction of the axis.* 

It is easily seen how the complete orbits, corre- 
sponding to the new direction of the axis can be 
constructed. (The points cG c lie on a circular arc 
about D, the middle point of S P.) 

Now, if the reader have carefully followed the pre- 
ceding reasoning_, he will readily see that a decrease of 
the normal force acting at the point M^, fig. 30, Plate 
VIII, will shift the other focus from H to 1; if the de- 
crease acts at Ma, the other focus will be shifted to 2 ; 
and so with the other points marked round the orbit, 
a decrease acting at Mg, M^^ M5, Mg, M7, or Mg, causing 
the focus to shift to 3, 4, 5, 6, 7, or 8, respectively. f 

* It is easily seen that the angle h P H, fig. 28, must be twice 
the angle T P ^, and h' P H equal to twice the angle T.P t'. To prove 
the first relation, we have the angle T P S equal to the angle T' P H ; 
and clearly, when T P T' assumes the position t P 1/, the angle 
T P S is less and the angle 1/ P H is greater than the angle T P S, 
by the angle t P T. Thus the angle ^ P H exceeds the angle t P S 
by twice the angle ^ P T ; and as 1/ P H is equal to the angle t P S, 
we have the angle HVh equal to twice the angle i P T. So with 
the other case. 

t The student will readily see why an oval shape is given to the 
curve through the points 1, 2, 3, .... 8. For the disturbing radial 
force, as is evident from fig. 17, is greater as the distance of the 
moon is greater. Thus it is greater when it acts at M5 than when 
it acts at Mi ; and, moreover, the moon is moving more slowly 
when at M5 than when at Mi. Each circumstance tends of itself 


Me Me 

Ibj 30. M?r?itaIJh7ve Inwa^rds. FySI. Mr-m/tljhrce outwards. 

llustra'tiTW the acH&n of tanaenZial forces aivarwus points ^aitorbil. 

taBZ. TanjfintiaJ 'AcceleraHon. r^^B. Jan^pTtlMl Retardation. 

^ Z9. IUustratm(^ the adi<nigfla7i^en:tml duturi)mfjo?'c^s. 
d.Froetor delf ~~~ 

THE moon's motions. 107 

It tlius appears that if the moon is anywhere on the 
arc Ml M3 M5, a decrease of the normal force causes 
the eccentricity to increase, the other focus being 
thrown farther from E, while if the moon is anywhere 
on the arc M5 M7 Mi_, a decrease of the normal force 
catises the eccentricity to decrease, the other focus 
being brought nearer to E. And again, if the moon 
is anywhere on the arc M7 Mj M3, a decrease of the 
normal force causes the perigee to regrede, while if 
she is anywhere on the arc M3 M5 M7, a decrease of 
the normal force causes the perigee to advance. The 
latter arc is considerably smaller than the former, 
but it is not described in a much shorter time, because 
it is at the apogeal end of the lunar orbit.* On the 

to make^H 5 greater than H 1, in the proportion of H M5 to H Mi, 
so that boih together would make H5 greater than Hi in the square 
of this proportion. But, on the other hand, there is a circumstance 
acting with contrary effect in such sort as to leave the increase 
only in the direct proportion of H M5 to H Mi, the fact, namely, 
that H M5, which sways, as it were, around the point M5, is 
shorter than H Mj. Thus, on the whole, H 5 exceeds H 1 in the 
proportion of H M5 to H Mj. In intermediate positions corres- 
ponding effects accrue, the disturbance of the other focus from the 
position H being always directly proportional to the distance of the 
moon from E. Hence the curve through the points 1, 2, 3, 4, 5, 
6, 7, and 8 is an ellipse resembling the ellipse Mi M^ M3 . . . . Mg, 
but placed as shown in the figure. Similar remarks apply when 
the radial force is increased. iSee fig. 31. 

* If in an ellipse having axes AC A', BOB', fig. 25, and foci S, H, 
we draw L H L' (the latus rectum) square to A A', and join S L, 
S L', the time in the arc L' A' L is to the time in the arc L A L' as 
the sectorial area L S L' to the remainder of the ellipse. Now if 
we draw L M, L' M' square to B B', it is obvious that the areas 
A S L, A S L' severally exceed the areas B C A, B' C A by the 


other hand, the normal action is considerably more 
effective when it is exerted on the moon at any point 
in the arc Mg M5 M7. Hence, since in the course of 
many lunar revolutions, a decrease of radial force 
must have acted at every part of the lunar orbit, there 
will be, on the whole, a balance in favour of those 
effects which accrue when the moon is near her 
apogee, or, so far as disturbances decreasing the 
radial force are concerned, the perigee will, on the 
whole, tend to advance, while (on the average of many 
lunar revolutions) the eccentricity will remain un- 
changed. Yet the eccentricity will not be constant, 
though it will undergo no permanent alteration. 

In like manner, an increase of the normal force, 
acting at the respective points M^, Mg, Mg . . . . Mg 
(fig. 31), causes the other focus to shift from H to the 
respective points 1, 2, 3, &c.* We see that if the 
moon is anywhere on the arc M^ M3 M5, the eccen- 
tricity is decreased by an increase of the normal 
force; while if the moon is anywhere on the arc 
M5 M7 Ml, an increase of the normal force causes 
the eccentricity to increase. Again, if the moon is 
anywhere on the arc M7 M^ Mg, an increase of the 

small spaces B M L, B' M' L', for the two triangles M F L, S F C 
are equal ; thus the area L S L' falls short of one half of the ellipse 
by the sum of the small areas B M L, B' M' L'. Hence the defect 
of the time in L' A' L from one half the periodic time bears to said 
time the small ratio which twice the area B M L bears to the whole 
area of the ellipse. 

* See note (f) at page 106. Exactly the same reasoning applies in 
this case as in the case there considered. 


THE moon's motions. 109 

normal force causes the perigee to advance, while if 
she is anywhere on the arc Mg Mg M7, the perigee is 
caused to regrede. And by reasoning precisely re- 
sembling that in the preceding paragraph, we see 
that on the average of many revolutions, during which 
increase of normal force must have acted on the moon 
at every part of her orbit, the perigee will have been 
caused to regrede by such increased radial action. 
Again also, the eccentricity, though not remaining 
constant, will undergo no permanent change through 
the change of the normal force. 

Now, since as we have shown (see fig. 19, Plate VI.) 
the sun's perturbing action tends, on the whole, to 
diminish the radial action on the moon, the maximum 
diminution being twice as great as the maximum 
increase, and diminution prevailing over a much 
larger portion of the moon's orbit, it follows that the 
balance of perigeal advance due to the diminution of 
the radial force, in any given number of revolutions, 
must exceed the balance of perigeal regression due 
to increase of the radial force. On the whole, then, 
the radial perturbations must leave a balance of 
perigeal advance. 

Next let us consider the tangential perturbing 
force. Here we have simpler preliminary considera- 
tions to deal with. We know that when the tangential 
force acts to accelerate the moon's motion, it tends to 
increase the major axis of the orbit, while when it 
retards the moon's motion, it tends to diminish the 
major axis. Now it is obvious that where there is no 

no THE moon's motions. 

change in the direction of the moon^s motion, the 
direction in which the other focus lies with respect to 
the moon cannot be altering (see note, p. 106), but 
that the other focus will be thrown farther away or 
brought nearer, according as the axis is increasing or 
diminishing, or, in other words, according as the 
tangential force is accelerative or retardative. 

This is illustrated for a particular position of the 
moving body, in fig. 29; but, as in the case of normal 
forces, the student would do well to repeat the con- 
struction for a variety of cases. 

In fig. 29, S is the attracting centre, round which 
the body is moving in the orbit AB'A'B, the other 
focus being at H. If, when the body is at P, its 
motion is accelerated, the other focus still lies towards 
H, but farther away than H, viz., at a position such 
as li, so that the new major axis, which, by the 
properties of the ellipse, is equal to the sum of the 
focal distances of P, is equal to S P and P h' together, 
or greater than S P and P H together. Then if we 
join S li , c , the middle point of S //, is the new 
centre of the orbit. We must take each of the lines 
c S a' and c li a, equal to the half of S P and P h' 
together; or, which is the same thing, we must 
make each of these lines c S a and c li af exceed 
CA' by half H /i'. The rest of the construction for 
determining the complete figure of the changed orbit 
is obvious.* 

* We draw b' c' h' square to a' a', and with centre // or S and 
distance a' c' describe a circular arc cuttinfj V c' h' In b', V, the 

THE moon's motions. Ill 

In like manner the construction proceeds, when the 
body is retarded at P, the new orbit having a focus at 
li and centre at c. The three positions of the centres 
at c, C, and c\ obviously lie on a straight line through 
D, the bisection of S P. 

Hence, when the tangential force is accelerative, 
fig. 32 shows the change in the position of the focus 
H for different positions of the moon in her orbit. 
When she is at M^, an acceleration shifts the farther 
focus to 1 ; when she is at Mg, the farther focus shifts 
to 2 ; and when she is at Mg, M^, Mg, &c., to Mg, the 
farther focus shifts to 3, 4, 5, &c., to S, respectively.* 

extremities of the new minor axis. It is to be noticed that the new 

eccentricity is not c' h' but -;; , ; so that it is not necessarily 

increased when the distance S h' is greater than S H. It can readily 
be shown that the eccentricity increases when the body is on the 
arc 6'a'b', and decreases when the body is on the arc b'a'^', 

* It will be easily seen that H 1 should be greater than H 5. 
For when the moon is in perigee the accelerative force is less than 
when she is in apogee, and acts for a shorter time, because she 
moves more quickly ; while, also, given accelerations increase the 
velocity in a smaller relative degree, since it is already large. 
Against this is to be set the circumstance that for a given increase 
of velocity the increase of the major axis is proportional to the 

velocity (as is easily shown from the relation = ), and 

therefore inversely proportional to the distances at apogee and 
perigee. There still remains an excess of increase of the major' 
axis when the moon is at her apogee. It is easily seen, also, that the 
maximum and minimum effects accrue when the moon is at her 
apogee and perigee respectively, the effects when she is at inter- 
mediate parts of her orbit being greater or less according as she is 
nearer to apogee or to perigee. 


Thus when the moon is on the arc M7 Mj M3, fig. 32, 
the distance of the other focus from S is increased by 
an accelerating tangential force ; whereas, if she is on 
the arc M3 M5 M7, the distance of the other focus 
from S is diminished. The actual eccentricity is 
increased or diminished, according as the body is on 
the arc B' M^ B or B M5 B'. Again, when the moon 
is on the arc M^ M3 M5, an accelerating tangential 
force causes the perigee to advance j while when she is 
on the arc M5 M7 M^, such a force causes the perigee 
to regrede. Thus, in any considerable number of re- 
volutions, tangential acceleration will cause (directly) 
neither advance nor regression of the perigee, the 
effects in one direction being, in the long run, exactly 
counterbalanced by those in the other. 

When the tangential force is retardative, fig. 33 
shows the change in the position of the focus H for 
different positions of the moon in her orbit. If she is 

at Mj, M2, Mg, M4, &c Mg, when the retardative 

action is exerted, the shift of the focus H is as towards 
1, 2, 3, &c., 7 and 8,* respectively. 

Thus we see that when the moon is on any part of 
the arc M7 M^ M3, the distance of the farther focus 
from S is diminished by retardative tangential force, 
while when she is on any part of the arc M3 M5 M7, 
the distance of the other focus from S is increased 
by such retardation. The actual eccentricity is 
diminished or increased accordiug as the body is on 
the arc B' Mi B or B Mg B'. Again, when she is on 
* See preceding note. 

THE moon's motions. 113 

the arc M1M3M01 retardation causes the perigee to 
regrede, while when she is on the arc M5M7M1 
retardation causes the perigee to advance. In this 
case, as in the former, the perigee neither advances nor 
regredes, on the average of many revolutions, so far as 
the direct action of the tangential force is concerned. 

It appears, then, that the radial disturbing force 
causes on the whole a progression of the perigee, 
while the tangential force does not directly produce 
any permanent eJBfect on the position of the perigee. 
It was to have been expected that a difference of this 
sort should assert itself, since fig. 22 shows that the 
radial disturbing force inwards is not equivalent to 
the outward disturbing radial force. On the other 
hand, the tangential force is altogether self-compensa- 
tory (see fig. 23), its action being alternately accelera- 
tive and retardative in the four quadrants, and equal 
in each, save on account of the eccentricity of the 
moon's orbit, which, however, favours permanently 
neither the accelerative nor the retardative eS'ect. 
This does not prevent, however, a temporary advance 
or recession of the perigee through the direct action 
of the tangential force, nor a temporary increase or 
diminution of eccentricity. 

But although the tangential force does not directly 
produce any permanent effect on the position of the 
perigee, it is indirectly as eff'ective as the radial force. 
In showing how this happens, I shall consider four 
special cases of the operation of the radial and tangen- 
tial forces in causing the advance of the perigee ; but 


114 THE moon's motions. 

I would invite tlie student who wishes really to master 
the subject to consider intermediate cases, carefully- 
making the requisite constructions and applying to 
them considerations resembling those which will now 
be applied to the four selected cases : 

Let the major axis of the lunar orbit (or, as it is 
called, the "line of apsides,") be directed as in fig. 34, 
Plate IX., towards the sun, the perigee being nearest 
to the sun as at p. Then the perturbing forces, for 
certain parts of the orbit, are indicated in the figure. 
At jp and a the radial force exerts its maximum out- 
ward actions ; at M and M4 it exerts its maximum 
inward actions. Near Oj, Og, O3, and O4, the tangential 
force has its maximum values.* 

Now, from fig. 30 combined with fig. 34, we see that 
the outward action of the radial force over the perigeal 
arc O4J9 Oi results in a regression of the perigee jf 

* The student should make tracings from figs. 34, 35, 36, and 37, 
and draw the radial and tangential resolved parts of the forces, 
precisely as in Plate VI. This will be found to be a most instructive 

t To prevent misconception I will go through the reasoning 
leading to this result, leaving to the student to deal in like 
manner with other cases as they arise. Fig. 34 shows that we 
have outward radial perturbing action as the moon is passing her 
perigee. Now, fig. 30 illustrates the eflPect of outward radial (or 
normal) perturbations. In this figure Mg Mi Ma is the perigeal 
arc, and we see that H is shifted towards 8 or 1 or 2, according as 
the body is at Mg or Mi or M2 when the perturbation takes place ; 
and in intermediate positions towards intermediate points. The 
perigee then is shifted from Mi backwards ; i. e. in the direction 
contrary to that indicated by the arrow on the orbit. With very 








[a / 


\ /I 







^llustratiTyf the Motion ^ the Feri^fe (f the Moon's Orbtl. 

H. Proctor deU 


while the reduction of the radial force over the apogeal 
arc O2 a O3 results in an advance of the perigee. It is 
obvious also that the inward action of the radial force 
over the arcs Oi Og and O3 O4 will have opposite and 
exactly counterbalancing effects. We have only to 
inquire then whether the regression just mentioned is 
less or greater than the advance. It is obvious that 
the outward action over the arc O4 _p Oi is less than 
the outward action over the arc Og O3 ; more- 
over^ the moon^ moving more swiftly in perigee than 
in apogee^ is exposed for a shorter time to the former 
smaller action than to the latter larger action. Ac- 
cordingly^ on both accounts^ the perigeal regression 
is less than the perigeal advance, so far as the radial 
force is concerned. There is therefore a balance of 
advance due to the radial action in a complete lunar 
evolution, when the perigee is as in fig. 34. 

Next as to the tangential action. In moving from b' to 
p, the moon is accelerated. Hence, from fig. 32 we see 
that the perigee recedes. In moving from p to h the 
moon is retarded; hence, from fig. 33 we see that the 
perigee still recedes. In like manner, in moving fro^ 
h to a the moon is accelerated ; hence, from fig. 32 the 
perigee advances. And in moving from a to b' the 
moon is retarded ; hence, from fig. 33 the perigee 
still advances. Now, in order to ascertain whether 

little practice the student will be able to deduce such results in an 
instant ; and with a little more he will have no occasion to refer to 
the figures 30, 31, 32, and 33. 

I 2 


the advance or recession is greater^ we have only to 
notice that, (1) the moon moves more rapidly over 
the arc p b than over the arc h a h\ so that any given 
acceleration or retardation will produce a smaller 
proportionate increase or decrease of velocity in the 
former than in the latter arc ; (2) fig. 34 shows that 
the actual forces are less in the former than in the 
latter arc; and (3) the forces act for a shorter time 
over the former than over the latter arc, because the 
moon moves over the former arc more quickly. On 
all three accounts the perigeal advance exceeds the 
perigeal recession. Thus there is a balance of advance 
due to the tangential force. But there is also a 
balance of advance due to the radial force. Hence, 
there is a total balance of advance when the moon is 
traversing her orbit placed as shown in fig. 34 . 

It is perfectly obvious that precisely the same result 
would have followed if the apogee had been turned 
directly towards the sun instead of the perigee. 

Next, let the major axis of the moon's orbit be 
placed at right angles to the line from the sun, as in 
fig. 35 ; and let similar constructions be employed in 
this case as in the former. Now, here it is obvious 
that the radial forces acting outwards on the moon as 
she traverses the arcs O4 b 0^ and O2 b' O3, produce 
opposite and exactly counterbalancing effects. But 
the radial forces acting inwards on the moon when 
traversing her apogeal arc 0^ a Og produce a regres- 
sion of the perigee (see fig. 31), which exceeds the 
advance of the perigee produced by the radial forces 


acting inwards on the moon when traversing her 
perigeal arc O3 p O4. For the former forces exceed 
the latter (see fig. 35)^ and act during a longer time, 
owing to the moon^s slower motion near her apogee. 
Hence, so far as the radial force is concerned, there is 
a balance of perigeal recession in a complete lunar 
revolution when the orbit is placed as in fig. 35. As 
regards the tangential force, there is retardation as 
the moon moves from M^ to a, and therefore (see fig. 33) 
the perigee recedes. In moving from a to M3, the 
moon is accelerated, and therefore (see fig. 32), the 
perigee still recedes ; thence to p there is retardation, 
and (see fig. 33) the perigee advances ; and, lastly, in 
moving to M^, the moon is accelerated, and therefore 
(see fig. 32) the perigee still advances. But the 
recession over the apogeal arc Mj a M3 is greater than 
the advance over the perigeal arc M3 jp Mj, for like 
reasons to those urged in the corresponding case in 
the preceding position. Hence, on the whole, there 
is a balance of perigeal recession due to the tangential 
force. But we have seen that there is also a balance 
of perigeal recession due to the radial force. Hence, 
there is a total balance of perigeal recession when the 
moon^s orbit is placed as shown in fig. 35. 

Obviously the same result would have been obtained 
if the major axis had been placed as in fig. 35, but 
with the positions of the perigee and apogee inter- 

Next, let us take such intermediate cases as are 
illustrated in figs. 36 and 37, where _p and a, as in the 



two former figures, indicate the position of the peri- 
gee and apogee respectively. In the case of fig. 36, 
as the moon is moving from ^ to h, the outward action 
of the radial force causes the perigee to recede (see 
fig. 30), while, as the moon is moving over the arc 
a V, the outward action of the radial force causes 
the perigee to advance. The advance exceeds the re- 
cession. Again, as the moon is moving over the 
arc h a, the action of the radial force (chiefly inward) 
causes the perigee to recede (see fig. 30) ; while, as 
the moon is moving over the arc b' ^, the chiefly 
inward action of the radial force causes the perigee to 
advance. The recession exceeds the advance. Thus 
there is a balance of recession to be set against the 
former balance of advance ; and though it is easily 
seen that in the actual circumstances indicated in 
fig. 36 the balance of advance is somewhat the greater, 
so that there remains a final balance of advance due to 
the radial force ; yet this balance is very small com- 
pared with the balances we had to deal with in the 
two preceding paragraphs. Again, as to the tangen- 
tial action. It is retardative over the arc M^ h Ma, 
and there (see fig. 33) causes recession; accelerative 
over the arc Mg a M^, and causes (fig. 32) an advance 
over Mg a, almost exactly compensated by recession 
over a Mg ; retardative over Mg V M4, and there' causes 
advance almost exactly compensating the recession 
over the arc M^ 6 Mg ; and, lastly, it is accelerative 
over the arc M4 p M^, and causes a recession over 
M4 Pj almost exactly compensated by advance over 


^ Mj. Hence, on the whole, tlie action of the tangen- 
tial forces produces no effect ; and, accordingly, when 
the moon's orbit is placed as in fig. 36, the combined 
action of the radial and tangential forces produces 
very little change in the position of the perigee. 

It will be found that precisely similar reasoning 
applies to the position of the lunar orbit illustrated in 
fig. 37. 

The perigeal advance attains its greatest rate when 
the lunar orbit has its major axis directed towards 
the sun (that is either as in fig. 34, or with the 
apogee nearest to the sun). The perigeal regression 
attains its greatest rate when the major axis is at 
right angles to the line from the sun (that is when 
the lunar orbit is either as in fig. 35, or with the major 
axis directly reversed). As the orbit changes from 
the former position to the latter (through the effect of 
the earth's motion round the sun), the advance of the. 
perigee per lunar month gradually diminishes until it 
vanishes, and then changes into regression, which con- 
tinually increases until it attains its maximum value. 
Thence as the orbit changes to the former position 
again, the regression gradually diminishes until it 
vanishes, and then changes into advance, which 
contiuually increases until it attains its maximum 
value. The maximum rate of perigeal advance is 
about 11 in a lunar revolution, the maximum rate 
of perigeal regression is about 9 in a lunar revolution. 
It is easy to see why the former exceeds the latter, 
for in the case illustrated by fig. 34, the advance of 


the perigee is due to the excess of a force which at 
its maximum is represented by a A.' , over a force 
which at its maximum is represented by jp A; while 
in the case illustrated by fig. 35, the regression of 
the perigee is due to the excess of a force which at 
its maximum is represented by the line E a over a 
force which at its maximum is represented by the 
line E _p. Clearly the former excess must be double 
the latter, just as the former forces are double the 
latter. It is easy also to see that the moon^s orbit in 
passing from the position where maximum advance 
prevails, to the position where maximum regression 
prevails, will be longer in a position involving ad- 
vance than in a position involving regression, simply 
because the degree of advance which is reduced to 
zero by such change of position, is greater than the 
degree of regression which is afterwards acquired. So 
that not only is the absolute maximum of perigeal 
advance greater than the maximum rate of perigeal 
regression, but advance continues during a longer 
period than regression. 

Another circumstance causes the advance to be 
greater than it would otherwise be. The balance of 
advance really depends on the excess of the disturbing 
action at and near apogee over the action at and 
near perigee. It will therefore be increased by any 
cause tending to increase the time during which the 
apogeal action takes place. Such a cause is to be found 
in the fact that the moon^s inotion round the earth 
does not exceed the sun's apparent motion round the 

THE moon's motions. 121 

earth, so greatly, when the moon is near apogee as 
when she is near perigee. Thus when near apogee 
the moon lingers longer than she otherwise would 
under those disturbing influences, which (on the 
whole) cause the advance of the perigee. We must 
not confuse this circumstance with what has been 
already mentioned respecting the effects of the moon's 
slower motion in apogee ; for when those effects were 
considered, the apparent motion of the sun was not 
taken into account. The sun's motion may be regarded 
as increasing the disproportion between the moon's 
motion in apogee and perigee. Thus let us repre- 
sent the moon's angular motion round the earth by 
14 when she is in perigee, and by 10 when she is in 
apogee; and the sun's apparent angular motion in 
the same direction by 1 . Then the apparent motion 
of the moon from the sun will clearly be represented 
by 13 when she is in perigee, and by 9 when she is 
in apogee. Thus the ratio of 14 to 10 is changed to 
the ratio of 13 to 9, which is larger than the former 
in the proportion of 65 to 63 ; and in about this 
proportion the perigeal advance is increased, owing 
to this cause. 

But another circumstance, the consideration of 
which will lead us to the recognition of the indirect 
effect of the tangential force already alluded to, is 
much more important in its effects. 

When the perigee is advancing, it is moving in 
the same direction as the sun around the earth; thus 
its angular displacement from the sun is due to the 


difference of these two advances. Since the sun^s 
mean advance during a lunar revolution is about 
27, and the maximum advance of the perigee in the 
same time is about 11, the displacement of the 
perigee from the sun may amount to so little as 
16 in a revolution. This happens when the perigee 
is advancing most rapidly, and tends to keep the 
perigee longer near that position with respect to the 
sun which is favourable to perigeal advance. Now 
when the perigee is regreding most rapidly, or at the 
rate of about 9 in a revolution, the displacement 
of the perigee from the sun is due to the sum of 
its regression and the sun's advance. It amounts 
therefore to 36 per revolution. Thus the perigee 
does not remain long in that position with respect 
to the sun which is favourable to perigeal regression. 
Hence the balance of perigeal advance is importantly 
increased. This reasoning is strengthened by the 
consideration that the rate of perigeal advance 
deduced from the maximum advance per month, is 
a less rate considerably than the rate deduced from 
the advance per hour, (say) when the apogee is 
advancing most rapidly (even in a month when on 
the whole the perigee regredes); and a similar con- 
sideration applies to the perigeal regression, (even in 
months, when, on the whole, the perigee advances). 

But it will be obvious that any cause which tends 
to encourage either the lingering of the lunar orbit 
in positions favourable to perigeal advance, or the 
rapidity with which it shifts from positions favourable 

THE moon's motions. 123 

to perigeal regression, must tend to increase the 
effect here considered, even though it might exercise 
no direct influence on the motion of the perigee. Now 
we have seen that when the moon's orbit is so placed 
that the radial force causes the perigee either to 
advance or recede, the tangential force causes the 
perigee to move in the same way, thus reinforcing 
the effects due to the radial disturbing action. 
These effects due to the tangential action do indeed 
counterpoise each other, so far as they are directly 
concerned ; in other words, their direct effect is nil in 
the long run. But insomuch as they reinforce those 
effects which (as we have seen) cause the lunar 
orbit to linger in positions favourable to perigeal 
advance, and to shift quickly from positions favour- 
able to perigeal regression, they indirectly reinforce 
the perigeal advance. Surprising though it may 
seem, these indirect results of the tangential action, 
these perturbations of perturbations as they may be 
called, actually exert so important an influence as to 
double the mean rate of perigeal advance. Newton 
either overlooked this indirect action, or rather fell 
into the mistake of supposing it might safely be 
neglected. Accordingly the only striking feature of 
the lunar perturbations which he was unable to explain 
in full, was the advance of the perigee. He could 
account but for about one-half of the advance. 
Clairaut, who first applied analytical investigations 
(as distinguished from Newton's geometrical method) 
to this question, deduced a result agreeing very 


closely with Newton's, having fallen into the same 
mistake. For some time it seemed as though, the 
theory of gravitation were endangered, Clairant himself 
suggesting that a force acting according to some 
other law than that of the inverse squares of the 
distances, seemed to be in operation. This opinion of 
a mathematician on a strictly mathematical question 
was energetically opposed by the non-mathematician 
BujQFon, who argued in favour of the simple Newtonian 
law. Clairaut was thus led to re-examine the sub- 
ject, taking into account considerations which he 
had hitherto neglected, and which he did not expect 
to find importantly influencing the result. To his sur- 
prise he found in the hitherto neglected indirect efiects 
of the tangential action, the explanation of the diffi- 
culty which had so long perplexed mathematicians. 
[See, however, note on p. 137.] 

The actual motion of the perigee from conjunction 
to conjunction with the sun is indicated in fig. 42, Plate 
XI., except that no account is taken of the oscillations 
which occur within the periods of successive lunations. 
The circle e^ e^ e.^, &c., is the orbit of the earth, while 
the lines jp^ e^ o^, p^ ^2 ^2^ Ps % ^sf indicate successive 
positions of the major axis of the moon's orbit, JPdJP^.jVzj 
&c., being the perigee. When the earth is at e^, the 
perigee is at p^, or in conjunction with the sun. Here 
the position corresponds to that illustrated in fig. 34, 
the advance is rapid, and accordingly we see that at 
the next station the perigee pa i^ ^^ longer parallel to 
the line e^ S, but has shifted in a direction agreeing 


with that indicated by the arrows on the moon^s 
orbit, that is_, in the direction of the moon^s advance. 
But when the earth has got to e^ (somewhat more 
than a quarter of a revolution) the position of the 
major axis corresponds to that illustrated in fig. 35, 
and the recession is rapid : hence in passing to this 
position from eg, and away from this position to e^, 
there is regression of the perigee, insomuch that the 
line ^4 64 a^ is shifted hack even beyond parallelism with 
'Pi 61 %. In the next two stages, however, there is cor- 
responding advance ; for at % the earth is so placed that 
the major axis is directed towards the sun : and we see 
that pc is very much advanced round the point eg. In the 
next two stages there is regression, so that jp^, though 
somewhat advanced as compared with f-y, is not so 
much advanced as pg. Lastly, as the earth passes to 
the position Cg, there is advance. Matters are now as 
at first ; and as the earth circles again round the 
sun, corresponding changes occur in the position of 
the perigee. The order of such changes is indicated 
in fig. 42 a, where E 1 represents the position of the 
perigee when at p-^ : we see how it advances to 2 (cor- 
responding to P2), recedes to 3 and 4, even behind 1; 
then advances to 5 and 6, recedes to 7 and 8, and 
lastly advances to 9. If we had begun with the 
position _p3 63 %, we should have had, in a complete 
circuit, the oscillatory progression indicated in fig. 42 h. 
It will be observed that the two figures agree per- 
fectly as respects the nature of the loops. Also the 
total angles of advance (1 E 9 in fig. 42 a, and 3 E 11 


in 42 h) are equal. Another circuit would take the 
perigee to the position E 17 in one case, and E 19 in 
the other. The arc of the advance between successive 
conjunctions of the perigee with the sun, that is the 
arc corresponding to e^ eg, fig. 42, is about 45 51 ^'_, the 
mean interval between such conjunctions being 41 1*767 
days. The perigee performs a complete circuit in a 
mean interval of 32 32 '5 75 days. 

Let it be particularly noticed that the point e^ is no 
definite point of the earth's orbit, but is taken to 
represent the earth's position at any epoch when the 
perigee is in conjunction with the sun. Again, it is 
to be noticed that the advance of the perigee does not 
take place in reality after the comparatively simple 
manner indicated in figs. 42 a and 42 6, since in each 
lunation there are two periods of advance and two of 
regression ; whereas these figures take into account 
only the balance of advance or regression. Moreover 
the periods of advance and regression are of various 
lengths in different lunations. 

Fig. 42 aptly illustrates the circumstances mentioned 
with respect to the lingering of the perigee in posi- 
tions favourable to advance. Let it be held so that 
S is on the right and pi e^ a^ on the left (or upside 
down), then the perigee is in the position favourable 
to advance, or as in fig. 34. Now let it be held with S on 
the right and p^ ^2 ^2 on the left. Then, since the earth 
has moved halfway of the way towards the position %, 
where the major axis is favourably placed for regres- 
sion, we should expect to have pa ^2 ^2 inclined half- 


way between tlie positions favourable for advance or 
regression, or as in fig. 36 : but instead of this we 
find Pa niucli nearer to the line joining S and e^. And 
so with the positions e^, e^, and e^. 

We have seen that the eccentricity of the lunar 
orbit, though affected during any given revolution, as 
well by the radial as by the tangential disturbing 
forces, is yet not subject to permanent alteration. 
The range within which the eccentricity varies is, 
however, one of the most important of all the features 
of the lunar theory. I do not propose here to enter 
on the consideration of the circumstances producing 
these oscillations in the value of the eccentricity, 
though the materials for the inquiry are contained 
in the considerations illustrated by the four figures 
30, 31, 32, 33, and those of Plate IX. The student 
will find no difficulty whatever in satisfying himself 
that when the axis is placed as in fig. 34 or 35, or 
in either of these positions, but with perigee and 
apogee interchanged, the eccentricity is not appre- 
ciably afi'ected in a complete lunation. When the 
axis is placed as in fig. 36 (or exactly reversed), the 
eccentricity is on the whole diminished in a complete 
lunation; and when the axis is placed as in fig. 37 
(or exactly reversed), the eccentricity is on the whole 
increased in a complete lunation. Thus it is easily 
seen that as the earth passes from the position e^ to 
the position eg (fig. 42), the eccentricity passes from 
a maximum to a minimum value ; at e^ it is again at 
a maximum ; at e^ again at a minimum ; and, lastly. 


at Cq it is at a maximum as at first. The range of the 
ecceutricity is so considerable as to exceed |ths of 
the mean value of the eccentricity. So that repre- 
senting this mean value as b, the maximum is about 
one-fifth greater, or 6 ; and the minimum about one- 
fifth less, or 4. Thus the greatest eccentricity exceeds 
the least in the proportion of 3 to 2. Since the mean 
eccentricity of the lunar orbit is 0*054908, the 
greatest and least values of the eccentricity are re- 
spectively about 0*066 and about 0'044. 

The irregularity of the perigeal motion and the 
variation of the eccentricity are oscillatory dis- 
turbances; and their combined influence on the 
actual position of the moon in her orbit is therefore 
also oscillatory in its efi'ects. It will be easily inferred 
that the moon's position is importantly modified at 
times by these causes, especially by the variation in 
the eccentricity, since the eccentricity causes the 
moon's motion in longitude to be unequal, and it is 
so much the more unequal as the eccentricity is 
greater. The moon, owing to these causes, may be 
in advance of, or behind, the place she would have if 
these perturbations had no existence, by no less than 
1 18'. This perturbation is called the evection, and 
is the only lunar perturbation which the ancient 
astronomers discovered. The discovery is commonly 
attributed to Ptolemy, though there are reasons for 
believing that it was actually made by Hipparchus. 

Hitherto we have supposed the lunar orbit to lie in 
the plane of the ecliptic, since we have regarded the 


three lines joining the snn and moon^ the sun and 
earthy and the earth and moon, as all lying in the 
plane of the moon's orbit. We know, however, that 
the moon's orbit is shghtly inclined to the plane of 
the ecliptic; and although the inclination does not 
importantly affect the v^ue of the radial and tan- 
gential forces, it produces a very important and 
interesting effect on the position of the lunar orbit. 
This effect we shall now proceed to examine. 

In the first place, let us take the general case of a 
body moving on a path inclined to any plane. Let 
N M N'j fig. 43, be part of the path of the body about 
the centre E, and let N m N' be the plane to which the 
motion is referred, so that N E N' is the line of nodes, 
and the angle P N m the inclination of the path. 
Then if, when at P, or passing from a node to its 
greatest distance from the plane of reference, the 
body is disturbed by a force acting toiuards that 
plane, it will proceed to move as along P k, the pro- 
longation of this new path (backwards) setting the 
node as at ??, or behind ^^^while the new inclination, 
or P n m, is obviously less than the former inclination 
P N m. It is equally clear that if the body is at Q 
when it is deflected towards the plane of reference, 
the new path placed as Q n, has its node n behind 
the former position N', but the inclination Q n' m 
is greater than the former inclination QN'm. 

Similarly if the disturbing force acts /rom the plane 
of reference and the body is at P, or anywhere on the 
arc N M (fig. 44), the node advances as to n, and the 



inclination increases ; wHle if the body is at Q, or 
anywhere on the arc M W, the node advances as to 
n' and the inclination diminishes. 

It will be observed that as the motion of the node 
is here referred to the direction of the body^s motion, 
the result applies equally whether N be the ascending 
or the descending node. 

We have, then, these general rules : Force towards 
the plane of reference, inclination diminishes while 
the body's distance from plane of reference is increas- 
ing, and vice versa ; but nodal line regredes through- 
out. Force /rom the plane of reference, inclination 
increases while the body's distance from the plane of 
reference is increasing, and vice versa ; but nodal line 
advances throughout. 

Now let us apply these results to the moon's motion 
round the earth, the plane of reference in this case 
being the ecliptic. 

Let us suppose^ first, that the line of the moon's 
nodes is placed (and remains during a complete revo- 
lution of the moon) as is shown in fig. 38, Plate X., N 
being the ascending node, and N' the descending node.* 
Then the line A' A, to points in which all the perturbing 
forces act, lies in the plane of the moon's orbit, being 
coincident with N W in direction and situation. It 

* The small lines surmounted by arrow-heads in this and suc- 
ceeding figures are intended to indicate the amount of the distance 
of the corresponding points of the lunar orbit, above or below the 
plane of the ecliptic, this last plane being supposed to be repre- 
sented by the plane of the paper. They are drawn to scale. 

rA^niTj j)l. 

ll/((stratu/(j //if Mofio/i of' (!((' AM's (jthe Noon's Or^//. 

KJ.Procior cMi 

THE moon's motions. 131 

is obvious^ then, tliat in this configuration these forces 
can exercise no efiect whatever in shifting the moon 
from the plane in which she is travelling at the time ; 
for each line of force lies in that plane.* The same 
will hold, of course, if the line of nodes has the same 
position but with the ascending and descending nodes 

Next, let us suppose that the line of nodes remains 
during a complete revolution in the position shown in 
fig. 39, N being, as before, the rising node, and N' 
the descending node. In this case E A lies below the 
plane of the moon's orbit, while E A! lies above that 

Since the moon in passing over the half-orbit 
N M N' is above the plane of the ecliptic, while the 
disturbing forces draw the moon towards points in 
the line E A below the plane of the lunar orbit, it is 
clear that throughout this part of the orbit the moon 
is drawn tovxirds the plane of the ecliptic. In like 
manner, since in passing over the half-orbit N' M' N 
the moon is below the plane of the ecliptic, while the 
forces draw her to points in the line E A' above the 
plane of the lunar orbit, it is clear that throughout 

* It is clear that, instead of resolving the force represented by 
such lines as M H, M' H' (figs. 20, 21, Plate VI.) into the tangential 
and radial forces only, we must begin (when we take the inclination 
of the moon's orbit into account) by dropping a perpendicular from H, 
&c., upon the plane of the moon's orbit, and taking this perpen- 
dicular to represent the force drawing the moon from her plane of 
motion, we must take a line from its foot to M M to represent the 
resultant of the tangential and radial forces. 
K 2 


this part of tlie orbit the moon is also drawn towards 
the plane of the ecliptic. Throughout the whole of 
her revolution, then, under the imagined condition, 
the moon is drawn towards the plane. 

Thus the line of nodes regredes throughout. The 
inclination diminishes while the moon is moving from 
N to M, increases as she moves to Wj decreases as 
she moves to M', and increases as she moves to N. 
It is therefore, on the whole, very little affected during 
the complete revolution. 

A like result obviously follows if the line of nodes 
is situated as in fig. 39, but the places of the nodes 

Thirdly, let the line of nodes be supposed to remain 
throughout a complete revolution of the moon in a 
position intermediate to those just considered, as at 
N W (figs. 40, 41). First let the moon pass a node in 
moving from M^ to M^, fig. 40. Then it will readily be 
seen (from considerations precisely like those in the 
preceding cases) that over the arcs M4 M^ W and 
Mg Mg N the moon is drawn towards the plane of the 
ecliptic, whereas over the two shorter arcs, N' Mg 
and N M4, she is drawn from that plane. Hence, on 
the whole, there is a balance of nodal regression in 
the complete revolution. I do not trace out the 
change of inclination, for the same reasons that I did 
not trace out the change of eccentricity, viz., first, to 
avoid prolixity, and secondly, because the change is 
an oscillatory one, producing no permanent effects. 
It will be found, however, that in the case indicated 


in fig. 40_, the inclination is on tlie whole undergoing 

Lastly, let the moon pass a node in moving from 
M4 to Ml, fig. 41. Then the node is regreding as the 
moon moves from N to M2, and from N' to M^, and ad- 
vancing throughout the rest of the moon^s motion ; 
hence, on the whole, the line of nodes regredes. The 
inclination decreases on the whole during a complete 
r evolution, t 

It is obvious that in all the four cases here con- 
sidered, matters will not be altered if the nodes be 
interchanged. J 

Thus we see that in all positions of the lunar orbit, 
except at the moment when the Hue of nodes is 
directed towards the sun, the nodal line regredes on 
the whole during each lunation, the regression being 
obviously most rapid when the line of nodes is at right 
angles to the line joining the earth and sun (or in the 
position shown in fig. 39). 

If the line of nodes remained fixed, it would be 
carried round the sun once in the year. But 

* It is only decreasing from M2 to M', and from M4 to M ; in- 
creasing everywhere else. The student will readily see this. 

t The inclination will be found to decrease everywhere except 
between M and M., and between M' and M4. 

X Excepting, of course, that there will be a slight change due to 
the fact that the sun's distance does not indefinitely exceed the 
moon's ; in other words, the perturbing forces on the side M4M1 Mg 
are slightly greater than those on the side M2 M3 M4. A similar re- 
servation applies to the corresponding cases in which the perigee 
and apogee were interchanged. 

134 THE moon's motions. 

under the actual circumstances tlie line of nodes is 
carried round in the manner represented in fig. 45, 
Plate XI. As the earth moves from e^, where the 
line of nodes 71^ e-^ n\ is directed towards the sun, 
the nodes regrede, slowly at first, and with alterna- 
tions of advance, but more rapidly, and with shorter 
periods of advance, until the earth is in position eg, 
not quite one-fourth of a revolution from e-^. At this 
time the line of nodes is square to the line from the 
sun. As the earth passes on to e^ the line of nodes still 
regredes on the whole, but with longer and longer 
alternations of advance, until when the earth is at e^ 
the line of nodes has the position % eg n\, or is again 
directed towards the sun. The regression recom- 
mences, and is continued with increasing effect to the 
position ey, and thence with diminishing effect to the 
position eg.* The actual nature of the nodal regres- 
sion, as the earth passes through the nine stages, 
^\i ^%) ^33 &c., is indicated in the figure 45, except 
that no account is here taken of the intermittence of 
the regression. Instead of a complete year being 
occupied by the moon's return in this way to a posi- 
tion where the rising node is again in conjunction 
with the sun as at first only 346" 60 7 days are so occu- 

* The inclination decreases as the earth moves from ei to e^ 
thence increases as the earth moves to eg, decreases as the earth 
moves to 67, and finally increases as the earth moves to e^. 

It is to be particularly noticed that e^ is not a fixed point on the 
earth's orbit, as the vernal equinox or the like. It is simply the 
point where, at the particular time illustrated, the nodal line n n' 
is directed towards the sun. 


RJ.Bve/07' deiL ""'^ 

136 THE moon's motions. 

the moon's line of nodes regredes, but that the regres- 
sion is most rapid when the plane of the lunar orbit is 
most inclined to the line from the sun. 

The plane's inclination to the ecliptic changes in 
an oscillatory manner in a mean period equal to one- 
half the interval of 346*6 days^ between successive 
returns of the line of nodes to such a position as is 
indicated in fig. 45 at ne-^ri and ne-^n, or in 173"3 
days. The variation amounts to about 8', by which 
the inclination is alternately greater and less than the 
mean value, rather less than 5 9^ 

Such are the chief perturbations to which the moon 
is subject. Others of lesser importance need not here 
occupy our attention^ because their discussion would 
introduce no new principles to our notice, at least 
none which could be discussed in such a work as the 

This chapter cannot properly be drawn to a con- 
clusion, however, without dwelling on the singular 
interest of the history of the researches made by astro- 
nomers into the subject of the lunar motions. The 
whole progress of the inquiry has been attended by 
difficulties only to be mastered by the most wonderful 
exercise of skill and patience. It was only the unique 
combination of powers possessed by Newton that per- 
mitted the problem to be grappled with in the first in- 
stance ; and even Newton would have failed but for 
certain fortunate circumstances by which he was 
assisted. Since his day the problem has been dealt 
with by the most acute mathematicians, by the most 

THE moon's motions. 137 

skilful observers. Mathematical analysis has been 
carried to an unhoped-for degree of perfection to 
account for peculiarities of lunar motion revealed by 
observation. Observation has been pushed to the 
utmost point of delicacy to detect peculiarities of 
lunar motion predicted by mathematical analysis. 
The history of the contest is adorned by the names 
of nearly all the leading observers and mathematicians 
of the last century and a half; Laplace and 
Lagrange; Euler, Clairaut^ and d'Alembert; Airy, 
Leverrier, Adams, and Cayley; Hansen, Delaunay, 
Peirce, and Newcomb ; a host, in fine, of names so 
distinguished, that it becomes almost invidious to 
particularize any among them. In the vs^hole 
history of the researches by vrhich men have endea- 
voured to master the secrets of nature, no chapter 
is more encouraging than that which relates to the 
interpretation of the lunar motions. 

Note. Since this chapter was in type, I have found that Prof. 
Grant, in an appendix to his " History of Physical Astronomy," 
records how Newton, in the original edition of the " Principia " 
(1687), gives very satisfactory values of the progression of the 
perigee in sizygy and its regression in quadrature. Thus he found 
11 21' for the monthly progression, and 8 1' for the monthly 
regression ; and a mean annual advance of 40. Modern tables 
assign 11, 9, and 40 40' 32'' for these quantities respectively. 
Newton refrained from publishing the details of his researches, but 
as Prof. Grant remarks, whatever Newton's method may have 
been, " it was manifestly one which was capable of grappHng with 
the main difficulties of the question." 




The moon^s motions in the heavens, as seen from the 
earth, are readily understood from what is known of 
her actual motions. I propose now to enter into a 
general consideration of these apparent motions of 
the moon, and of the varying aspect which she accord- 
ingly presents to us. It would be possible to fill a 
much larger volume than the present with the detailed 
discussion of these matters ; nor would such a volume 
be wanting in interest, at least to those having 
mathematical tastes. I do not indeed know of any 
subject which a geometrician could better wish to 
examine. It is full of neat and interesting problems, 
and might worthily occupy many years of labour. 
But in this volume such researches would be out of 
place. We must be content with such a consideration 
of the subject as shall leave none of its salient features 
unexplained. In passing it may be remarked that 
even such a treatment of the moon's apparent motions 
has long been a desideratum, inasmuch as our text- 


books of astronomy have hitherto left these matters 
almost untouched. 

In the first place, then, it is to be noticed that the 
moon completes the circuit of the heavens on the 
average in 27-322 days, that is in 27^- 7^- 43' 7^- If 
we watched her motion from the time when she was 
in conjunction with any given star until the next 
conjunction, and the next again, and so on, for many 
successive conjunctions, we should find that the mean 
interval is that just stated. This is called the sidereal 

If, however, instead of taking a star, we took the 
point on the heavens where the ecliptic crosses to the 
north of the equator, we should not find the interval 
exactly the same as the sidereal month ; because this 
point on the heavens is constantly, though slowly, 
moving backwards, or so as to meet the moon's motion. 
This point called, as all know, " the first point of 
Aries" makes the complete circuit of the heavens 
in 25,868 years; and therefore in a sidereal month 
travels over a very minute arc indeed, less in fact 
than 4". So that the difi'erence between this new 
kind of month, called the tropical month, and the 
sidereal month, is very minute. The mean tropical 
month is necessarily slightly less than the sidereal. 
The latter is, with great exactness, 27*32166 days, 
the tropical month is 27*32156 days, or about 6^ 
seconds shorter. 

Now let us in the first instance consider this motion 
as though it took place in the ecliptic, and uniformly. 


SO that in fact we are supposing the moon to move 
apparently in the same course among the stars as the 
sun, only that instead of taking about 3665 days in 
completing the circuit she takes about 27^ days. 

Let E E', fig. 46, Plate XII., represent a part of the 
earth^s path round the sun S, and let M^ Mg Mg M^ 
be the path of the moon, and suppose that the 
moon is at Mj when the earth is at E. Then it is the 
time of '^ new moon ; " the moon lies towards the 
sun^s place, and if she could be seen, would be at the 
same part of the ecliptic, or in conjunction with the 
same star s. Let E E' be the arc traversed by the 
earth in 27*322 days, or in a sidereal month. Then 
the moon has gone once round, and is in conjunction 
with the same star, in other words, the line Wrrii / 
directed towards the moon, is in the same direction 
as E s, that is, E' m^ / is parallel to E s. But the 
moon has not come up to the line E^M/S, joining the 
sun and earth. Some time has still to elapse, there- 
fore, before it is again new moon. In like manner, if 
the moon had been at Mg when the earth was at E, 
it was the time of "first quarter,^' she would be at m^ 
when the earth is at E', in other words, she would 
not yet have reached M^, the place of " first quarter." 
And similarly if it had been " full moon,^' " third 
quarter,'^ or any other lunar epoch, when the earth 
was at E, the corresponding epoch would not have 
arrived, when a sidereal month had elapsed. 

We see then that the lunation, or the time in which 
the moon goes through her phases, is longer than the 




to ahorU /OO U?)ie.s 
^^^ dislance M^Mg. M: 


) \ 4 C 



Illfist/'aUn// the ^Moon's .yipparent 'AfoUom\F/fa6'es,i^o. 


sidereal or than tlie tropical montli. And it is very 
easy to calculate tlie exact length of the lunation, or, 
as it is called, the synodical month. In 27*322 days, 
the moon has not completed the whole cycle of her 
phases, but only the portion M/ M3' m^ out of the 
whole cycle, that is, she has completed the whole 
cycle, less the portion m^ M^\ Now, the angle 
rrii E' M/ is obviously the same as the angle E' S E ; 
hence the part wanting from the complete cycle bears 
to the Vv^hole cycle the same ratio that E E' bears to 
the complete orbit of the earth, or that 27*322 days 
bears to 365*242 days. The moon, then, in 27*322 
days, completes only i^Sths of a lunation (the 
numerator being obtained by taking 27*322 from 
365*242). So that a mean synodical month exceeds 
a mean sidereal month (or 27*322 days) in the same 
proportion that 365*242 exceeds 337*920. Increasing 
27*322 in this proportion (a mere rule-of- three sum), 
we obtain 29*531,* which is the length of a lunation. 

The phases of the moon are explained in text-books 
of astronomy. But a few remarks on the subject may 
be useful. 

Let Ml M2 M3 . . . . fig. 47, Plate XII., represent 
the moon^s orbit, the sun being at S, only many times 
farther away than in the figure. The earth and moon 
are relatively much exaggerated in dimensions ; and 
the moon is shown in eight equidistant positions, as 
though she performed a complete circuit, while the 

* More exact values are given in the tables. 

142 THE moon's changes 

earth remained at E. Now obviously, when the moon 
is at Mj, her darkened side is turned towards the 
earth, and she cannot be seen. She is as at 1, fig. 48. 
As she advances towards Mg, the observer on 
the earth B, and supposed to be standing on the 
half of the earth shown in the figure, sees the moon 
on the left of the sun, that is, towards the east,* 
and he would clearly see the right or western side of 
the moon partly illuminated. The case, so far as this 
illumination is concerned, is exactly the same as 
though the moon at Mj had turned an eighth round on 
an axis upright to the plane of her motion, in such a 
way as to bring into view the parts beyond her eastern 
edge. Thus, the aspect of the moon is as shown at 
2, fig. 48. It is readily seen that when she is at 
M3, fig. 47, her aspect is as at 3, fig. 48 ; and so on. 

All this is as explained in the text-books. But 
there are two points, even in this elementary matter, 
which may need a word or two of explanation. 

First, as to the position of the lunar crescent. We 
see the moon in varying positions on the sky ; and at 
first sight there appears to be no definite relation 
between her position and the position of her cusps or 
horns. Indeed, this feature of her aspect has seemed so 
changeful and capricious that it has even been regarded 
as a weather-token. In reality, however, there is a 
simple relation always fulfilled by the moon^s cusps. 
The line joining them is always at right angles to the 

* The reader should here hold the plate so as to have E towards 
him, and S and M2 from him. 


great circle passing through the sun and moon.* As 
the moon is always near the ecliptic, this amounts to 
saying that the line joining the cusps is always nearly 
at right angles to the ecliptic. It follows, of course, 
that as the angle at which the ecliptic is inclined to 
the horizon is variable, so the position of the line 
joining the cusps varies with respect to the horizon. 
As respects the gibbous moon (or moon more than 
half-full), these variations are not much noticed; but 
in the case of the crescent moon, generally observed 
rather near the horizon, they are very noteworthy. 
For instance, let the time of year be such that the 
part of the ecliptic near the western horizon, soon 
after sunset, is inclined at nearly the greatest possible 
angle to the horizon, that is, let the season some- 
what precede the vernal equinox, the time, as we 
know, when the zodiacal light is most conspicuous in 
the evening. Then in our latitudes, the inclination 
of the ecliptic to the horizon is about sixty-two 
degrees, and supposing the moon on the ecliptic, and 
young, as shown at Mj, in fig. 49, Plate XII., the line 

* This will perhaps seem obvious to most readers. The proof 
of the proposition is comprehended in the following considerations : 
The circle bounding the illuminated half of the moon necessarily 
has its plane at right angles to the line joining the centres of the 
sun and moon ; the circle bounding the moon's visible hemisphere 
necessarily has its plane at right angles to the line joining the 
centres of the earth and moon : thus the intersection of these circles 
or the lunar cusps, must lie on a line at right angles to the plane 
containing the three centres, that is, to the plane of the great 
circle through the sun and moon. 

144 THE moon's changes 

ioining the cusps will only be inclined about twenty- 
eight degrees to the horizon. But next_, suppose that 
the moon at this time is at her greatest distance 
north of the ecliptic, or at Mg, five degrees from the 
position Ml, and about the same distance as in the 
former case from the sun. Then the great arc- circle 
S M2 from the sun to the moon is inclined ten or more 
degrees (according to the moon's age) to S Mi, and 
the line joining the cusps is, in this case, inclined less 
than 18 to the horizon. Indeed, when the moon is 
very young, the angle MiS Mg is considerable. Hence 
S Mg makes a considerably larger angle with the 
horizon than S Mi, and the line joining the cusps is, 
as shown in the figure, much more nearly horizontal. 
A very young moon seen soon after sunset, under 
these circumstances, may have the line joining its 
cusps quite horizontal, or even have the northern 
cusp lower than the southern.* Like considerations 
apply to the case of the old crescent raoon,t before 
sunrise, soon after the autumnal equinox. 

Next, however, suppose the western part of the 

* It is hardly necessary to say that the exact angles for any 
position can be quite readily calculated ; but the matter is not of 
such a nature as to require the introduction of such calculations 
here. The student acquainted with the elements of spherical tri- 
gonometry may find interesting and not nninstructive occupation 
for a leisure hour or so in considering a few cases. The angle 
M2 S Ml is more than 10 when the moon is less than one-eighth 
full, or halfway to the first quarter. 

t The word crescent here means merely crescent-shaped, not 
crescent in the sense of increasing. 


ecliptic, at its least inclination to the horizon, soon 
after sunset, or the time of year shortly before the 
autumnal equinox. The state of things is that illus- 
trated in fig. 50, Plate XII. Then in our latitudes 
the inclination of the ecliptic to the horizon is about 
15 degrees, and supposing the moon on the ecliptic 
and young, as at Mg, the line joining the cusps will be 
inclined about 75 degrees to the horizon. But sup- 
pose the moon, as at M4, at her greatest distance 
south of the ecliptic, or five degrees from the position 
Mg, and about the same distance from the sun, then 
the great circle S M4 from the sun to the moon is in- 
clined more than ten degrees to S Mg ; and the line 
joining the cusps may be much more nearly upright 
than when the moon is as at Mg. But this line 
cannot be actually upright when the sun is below the 
horizon, for the line must always be square to the 
great circle through the sun and moon, and, of course, 
when the moon is above and the sun below the horizon, 
this great circle is inclined to the horizon, and a line 
perpendicular to it is correspondingly inclined from 
the vertical. Similar considerations apply to the case 
of the old crescent moon before sunrise, soon after 
the vernal equinox. 

We see, from these extreme cases, that the line 
joining the moon's cusps can have every inclination, 
from being nearly vertical to a horizontal position, 
and even that the northern cusp may be below the 
southern, according to the season of the year and the 
moon^s position in her orbit. So that, to assert that 



there will be such and such weather when the line 
joining the cusps is seen (for instance) nearly hori- 
zontal, the moon being new, is the same as asserting 
that there must be such and such weather at the time 
of new moon in February and March, if the moon is 
then nearly at her maximum distance from the 
ecliptic. And so with all such cases. If there were 
any value at all in such predictions, they would 
imply the strictly cyclic return of such and such 

Secondly, as to the rate at which the moon changes 
in shape. 

Let us suppose that ABC D, fig. 61, Plate XII., 
represents the moon^s disc (dark in the first instance), 
and that when the illumination begins on the right, B D 
is the line joiniug the cusps. Now, from what has just 
been shown, it is seen that the position of B D must 
vary during the progress of the lunar montb, unless 
we suppose the moon to be moving in the ecliptic. 
As, however, we may wish to know the rate at which 
the moon Jills, we may make this assumption for 
convenience. Now, the variation of phase obviously 
corresponds exactly to the supposition that the semi- 
circle BAD, which separates on the right the dark 
from the illuminated hemispheres, rotates round the 
axis B D, the point A travelling apparently straight 
across to C, but in reality, of course, traversing a 
semicircle, which is seen projected into the straight 
line A C. Now, to find what point of A C will have 
been reached by the advancing boundary of the 


illuminated hemisphere, we have only to imagine a 
point traversing the semicircle ABC uniformly in 
14| days. From whatever position as a, this moving 
point would have reached in so many days, we must 
let fall a perpendicular am on A 0. Then w will 
obviously be the position of the advancing edge at the 
time in question ; for A m is obviously the projected 
view of an arc exactly equal to A a. Hence the 
semi-ellipse B m D indicates the concave outline of 
the illuminated portion at this epoch. Thus, in the 
figure, A a is one-fourth of the semicircle ABC, and, 
therefore, A B 7?! 1) is the shape of the moon's crescent 
when she is an eighth of a lunation old, or nearly 
3/o days old. In like manner, if h be midway between 
B and G, hn perpendicular to A C gives us B n D, 
the elliptical outline of the gibbous moon, at the time 
when she is gibbous, midway between first quarter 
and full ; and A B n D is the phase of the moon at 
this time, when she is about lly"^ days old. It is readily 
seen that B C D m is the figure of the gibbous moon 
at a time midway between ''full'' and third quarter; 
while, lastly, B C D % is the figure of the waning 
moon at a time midway between third quarter and 

Now, as the lunar month contains about 29^ days, 
if we divide A C into 14| equal parts, as shown by 
the numbered division-lines, we obtain, by letting fall 
perpendiculars, the daily progress of the advancing 
rim of light from new to full, as shown by the 
numbered division-marks on C A. We have only to 
L 2 


invert the figure to have the daily progress of the re- 
ceding rim of light from '^ full " to "t\.qw." Or we may 
construct such a figure on a larger scale, and divide 
the semicircle ABC into 59 equal parts, then the fact 
of perpendiculars let fall from the division-points 
upon A C will correspond very nearly indeed to the 
progression and retreat of the advancing illuminated 
rim from " new " to " full/^ and thence to " new " 
again, at six-hourly intervals. 

Let us next consider the actual motions of the 
moon in the heavens at different times. We shall 
have, in so doing, to take into account the inclination 
of the moon^s path to the ecliptic, as well as the 
eccentricity of the lunar orbit. 

So long as we regard the moon as moving in the 
ecliptic, we can at once determine the nature of 
the moon^s movements during any month of the 
year, by considering where the sun is placed on 
the ecliptic during that month. Thus in March the 
sun crosses the equator ascendingly. Hence, at the 
time of new moon, the moon is near the equator, and, 
like the sun, is about as many hours above as below 
the horizon. As the moon passes to the first quarter, 
she traverses the ascending part of the ecliptic, and 
at the time of first quarter is near the place occupied 
by the sun at the midsummer solstice. In other 
words (for we cannot too directly refer these motions 
to the stellar heavens) the moon is near the place 
where the constellations Taurus and Gemini meet 
together. Thus the first-quarter moon in spring is a 


long time above the horizon, and is high when in the 
south, Hke the sun in midsummer. She passes on to 
full, when she is again near the equator, or rather 
when she is "fulP^ in March (which may be earlier 
than the date when she is at her first quarter) she is 
near the equator where the ecliptic crosses it, or in 
Virgo. So that the full moon in spring is about 
twelve hours above the horizon, and as high when 
due south as the sun in spring. The "third- quarter 
moon'' in March is, in like manner, nearly in the 
part of the ecliptic occupied by the sun in winter, or 
where the ecliptic crosses the equator in Sagittarius. 
She is therefore but a short time above the horizon, 
and low down when due south, like the winter sun. 
And it is easily seen how at intermediate phases she 
occupies intermediate positions. 

By similar reasoning, we find that in midsummer 
(i) the new moon is in Taurus or Gemini,* and long 
above the horizon ; (ii) the first-quarter moon is in 
Virgo, and about twelve hours above the horizon; 
(iii) the full moon in Sagittarius, and a short time 
above the horizon; (iv) the third-quarter moon in 
Pisces, and about twelve hours above the horizon. 
In mid-autumn, (i) the new moon is in Virgo, and 
about twelve hours above the horizon ; (ii) the first- 
quarter moon, in Scorpio or Sagittarius, and only a 
short time above the horizon ; (iii) the full moon in 
Pisces, and about twelve hours above the horizon; 

* The reference throughout is to the constellations, not to the 


(iv) the three-quarter moon in Taurus or Gemini, and 
a long time above the horizon. And, lastly, in mid- 
winter (i) the new moon is in Scorpio or Sagittarius, 
and only a short time above the horizon; (ii) the first- 
quarter moon in Pisces, and about twelve hours above 
the horizon ; (iii) the fall moon in Taurus or Gemini, 
and along time above the horizon; and (iv) the third- 
quarter moon in Virgo, and about twelve hours above 
the horizon. 

The student will find no difficulty whatever in 
extending these considerations to other months, or in 
applying much more exact considerations to special 
cases. For he will notice that what has just been 
stated presents only the rougher features of the 
matter. But nothing can be easier than to apply 
the first rough corrections for such an inquiry. 
Supposing, for example, that we wish to know 
generally what will be the hnoon's diurnal path 
(that is her course round the heavens during the 
twenty-four hours) when she is at her first quarter 
on the 10th of April: we know that on the 10th of 
April the sun is some twenty degrees past the vernal 
equinox, which he had crossed on or about the 20th of 
March; the moon at her first quarter is 90 farther 
forward, or some twenty degrees past the place of the 
summer solstice; corresponding to a position on the 
ecliptic, about equidistant from the two stars k and 
S Geminorum. Her course above the horizon will 
correspond to the sun^s course about twenty-one 
days after the summer solstice, that is, on or about 


July 11th.* Similarly any other case can be dealt 

Before passing from this part of our subject, we 
may here conveniently consider the phenomena of the 
harvest moon and of the hunter's moon. 

If the moon moved in the equator, she would rise 
later night after night by a nearly constant interval ; 
or, in other words, the actual number of hours be- 
tween successive risings (or settings) would be con- 
stant. But as she moves on a path considerably 
inclined to the equator, this does not happen with her 
any more than it does with the sun ; moreover, as she 
moves much more rapidly along the equator than the 
sun does, the difference is much more perceptible. If 
we consider two .extreme cases, we shall see the 
reason of this. Let H H', fig. 52, Plate XII., be a 
portion of the eastern horizon, E the true east point, 
E Q the equator ; and let us suppose that when the 
moon rises on a certain night she is on the equator at 
E. She is then carried by the diurnal motion along 
E Q to her culmination in the south, and so to her 
setting place in the west. Now if her orbital motion 
were on the equator, she would be on the next night 
at the same hour at a point such as m on the equator 
(Em being an arc of about 12 12"), and would be carried 

* In a work now out of print, called the " Constellation Seasons," 
I introduced a map showing the sun's diurnal course at different 
dates, in such sort that his elevation and bearing at any time could 
be at once ascertained. Such a map serves many useful purposes 
besides those for which it is primarily intended. 


by the diurnal motion to E, where she would rise 
about 50| minutes later than on the former day (and 
about 13 in advance of her former place). But her 
actual motion is nearly on the ecliptic ; and when she 
was at E on the first day the ecliptic must have been 
^n one of the two positions eE or e'E. (In other 
words, E must be the point where the ecliptic crosses 
the equator, either descendingly or ascendingly.)* 
Now in the former case, the moon on the second 
night will be as at M, and will be carried by the 
diurnal motion to the point h on the horizon ; in the 
latter she will be as at M', and will be carried to 
the point fi ; and obviously M ^ is a much longer arc 
than M' }i\ In fact, if K E K' be part of the equinoctial 
colure (or circle square to the equator through the 
equinoctial point E), the two arcs MK and M'K' are 
obviously equal,t and we see that M h exceeds, while 
M' li falls short of the common length of those equal 
arcs by the very appreciable equal arcs K li and K' hf. 
Thus the hour of rising in the former case will be 
later than in the latter, by the time corresponding to 
twice the diurnal arc K h or K' K, as well as by a not 
inconsiderable increment of time due to the fact that 
the moon is all the while moving on her orbit, and 
moves farther, of course, the longer she is delayed. 
The hour of rising will in both cases be later than the 

* The direction in which we follow the ecliptic is contrary to 
that of the diurnal motion, because the sun's annual motion in the 
ecliptic is from west to east. 

t They are also each very nearly equal to E m. 


hour at which the moon rose on the preceding night 
(at least in our latitudes, and everywhere save in very- 
high latitudes), but the diflference will be much greater 
in one case than in the other. 

Now these are the extreme cases : the ecliptic can 
never cross the horizon at a greater angle than e E H', 
or at a less angle than e'EH'. Accordingly still 
assuming that the moon moves in the ecliptic wo 
shall have the greatest possible difference between 
the hours of rising when the moon is on the ecliptic 
placed as at e E M, and the least possible difference 
when she is on the ecliptic placed as e' E M' ; and if 
the moon is " full " or nearly so, when in one or other 
of these positions, the peculiarity will be very note- 
worthy. In one case, we shall have a remarkable 
retardation in the hours of rising on successive days, 
and in the other as remarkably small a difference. 
Now the full moon is in or near the former position in 
spring, for then the new moon is, with the sun, at or 
near the ascending node of the ecliptic, and therefore 
the full moon at or near the descending node. Ac- 
cordingly in spring the difference between the hours 
at which the full moon rises on successive nights is 
considerable. It amounts, in fact, on the average, in 
our latitudes to about an hour and twenty minutes,* 

* There is a tahle in Ferguson's Astronomy which seems to 
imply differently, since he gets Ih. 16 m. as the greatest possible 
difference between the hours of successive rising or setting of the 
moon, when the inclination of her orbit to the ecliptic is taken into 
account ; and this value has been carefully reproduced in our text- 


the mean interval being only about 50^ minutes. 
And the full moon is near the ascending node of the 

books of astronomy. But it should be noticed that Ferguson did 
not compute the values in this table, but only estimated the values 
" as near as could be done from a common globe, on which the 
moon's orbit was delineated with a black-lead pencil," and he was 
not successful even in his application of this very rough method, 
by which, or by a simple method of projection, it may readily be 
shown that the maximum difference is greater and the minimum 
dijfference less than Ferguson supposed. If the eccentricity of the 
moon's orbit and her consequently variable motion be taken into 
account, a yet greater difference results. It is easy to obtain equa- 
tions whence we can calculate the diflference in the hour of rising 
under the circumstances in question. They are as follows, the 
assumption being made that the moon is crossing the equator at 
rising : Let a be the inclination of the moon's path to the equator 
(a ranging in value between 28 44' and 18 10'), I the latitude of 
the station. Then let h be the moon's mean hourly motion on the 
ecliptic (about 30^ minutes of arc), x the time in hours between 
her rising on the day when she is on the equator and on the next 
day. Then her motion on the ecliptic is x h. Put x h = 9. 
Take then 

sin ;// = sin a sin (i) 

and sin ^ = tan I tan ;// (ii) 

Then is approximately the hour-angle by which the interval 
between successive risings exceeds or falls short of the mean 
interval (1 d. 5()| m.). So that 

X = 24-84 ~; that is = 24-84 /i ^ 

= 373 <p approximately. 

These equations are theoretically sufficient to determine (or x) ; 
but practically, it is sufficient to adopt a value of ,- (half an hour is 
near enough), giving x = 24*34 ; 9 = 12 22^' about. Then use (i) 
and (ii) to calculate <p, and repeat the process, using in it the value 
of <p thus deduced. 


ecliptic in autumn^ for then the new moon is^ with 
the sun^ at or near the descending node of the ecliptic. 
Accordingly, in autumn, the difference between the 
hours at which the full moon rises on successive 
nights is small. It amounts, in fact, on the average, 
in our latitudes to rather more than twenty minutes 
(or about half an hour less than the mean interval). 

But the inclination of the moon^s orbit and the 
moon's variable motion due to the eccentricity of her 
orbit cause these results to be considerably modified. 
We can at once consider this feature (proposing pre- 
sently to discuss more particularly the moon's motion 
on her inclined eccentric orbit). Let us suppose that 
when at E, fig. 52, Plate XII., the moon is crossing 
the equator, ascendingly or towards M', and is also at 
the rising node of her orbit. Then, instead of following 
the course E M', she will travel along such a course as 
is shown by the dotted line E 1, or will be yet nearer 
than M' to the horizon at the end of the twenty-four 
hours, in other words, the interval between succes- 
sive risings at this season will be yet more shortened 
than we have found it to be on the assumption that 
the moon moves on the ecliptic. In like manner 
if when at E, and crossing the equator descendingly, 
the moon is at her descending node (which will 
obviously correspond to the period when she crosses 
the equator ascendingly while near her ascending 
node) then, instead of following the course E M, 
she will follow the course E 4, or will be yet 
farther than M from the horizon at the end of the 

156 THE moon's changes 

twenty-four hours, in other words, the interval be- 
tween successive risings will be yet further lengthened 
than we have found it to be on the assumption that 
the moon moved in the ecliptic. On the contrary, if 
the moon, when crossing the ecliptic ascendingly, is 
at her descending node (so following the course E 2), 
while when crossing the ecliptic descendingly she is 
at her ascending node (so following the course E 3), 
the intervals between successive risings and settings 
will be less markedly affected than on the assumption 
that the moon moves in the ecliptic. These are the 
extreme cases either way. It is readily seen, how- 
ever, that the position of the moon as to the perigee 
and apogee of her orbit must also have an effect, 
since her motion from E will be greater or less 
according as she is nearer or farther from her perigee, 
and the interval between successive risings will be 
diminished or increased respectively. 

Taking all these considerations into account, it is 
found that instead of the moon rising about 20 
minutes later night after night for several successive 
days at the time of harvest moon, she at times rises only 
nine or ten minutes later on successive nights ; while 
at other times, at the same season, the difference 
exceeds half an hour. As regards the maximum dif- 
ference between the hours of rising of the full moon 
in spring, it varies from about an hour and ten 
minutes to about an hour and a half. 

It is to be noticed that in every lunation corre- 
spondiug variations occur, because the moon neces- 


sarily passes through Pisces and Aries, and through 
Yirgo and Libra in each lunation. But it is only in 
spring that the full moon is in Libra and Virgo, and 
in autumn that the full moon is in Pisces and Aries. 
The autumn phenomena are the more important, since 
they result in nights almost completely moonlit for 
four or five days in succession. We have at, and near 
the time of full moon in September, the moon rising 
not far on either side of six in the evening, and 
though the hour of setting varies considerably, yet 
this is obviously a matter of small importance, since 
the moon sets in the morning hours. The operations 
of harvesting can thus be continued far on into the 
night, or all night if need be. This relates, however 
(at least in England), to the full moon preceding the 
middle of September, for harvesting operations are 
nearly always completed throughout England before 
that time. The full moon following September, which 
partakes to about an equal degree with that preceding 
the autumnal equinox, in the peculiarity we have been 
dealing with, is sometimes called the hunter's moon. 

In latitudes higher than ours the phenomena of the 
harvest moon and hunter's moon are more marked, 
because the angle HEM' (fig. 52) grows smaller and 
smaller as the arctic circle is approached. At the 
arctic circle this angle vanishes, and the moon, when 
moving parallel to the ecliptic, rises night after night 
(for a time in each lunation) at the same sidereal 
time, or nearly four minutes earlier on successive nights. 
However, into such peculiarities as these we do not 


here enter, because the subject would thus become 
an exceedingly wide one, while in reality there is 
little importance in the relations thus involved, since 
in the arctic regions there are no harvesters to be 
benefited, nor is hunting there pursued in the night 

But we must now take into account the circum- 
stance that the moon moves on an orbit somewhat 
inclined to the ecliptic. It will, in the first place, be 
manifest that if the position^ of the plane in which 

^ I use this word to indicate not the actual place of the plane 
in question, but the manner in which it is posed in space. Thus 
the position of the earth's equator-plane would, according to this 
usage of the word, be described as identical (neglecting precession) 
throughout the year, the position of the earth's orbit-plane identical 
year after year as the sun moves onward with his family of dependent 
orbs through space, the position of the plane of the Saturnian rings 
identical throughout the Saturnian year, and so on. A discussion 
occurred a year or two ago, in the pages of a weekly journal, as to 
the proper word to indicate this particular relation, and I advocated 
then the use of the word " position " as on the whole the most suit- 
able. The question is one to which my attention has been particu- 
larly drawn, because it has chanced that repeatedly in my writings 
I have had to deal with this feature ; and I have found no word so 
readily understood in this particular sense as the word " position." 
At the same time I must admit, first, that the word is not wholly 
free from objection, and secondly, that several mathematicians, to 
whose opinion I feel bound to attach great weight, are opposed to 
its use in this sense. Unfortunately they suggest no other term. 
It appears to me that the objections to the use of the word " posi- 
tion " in the sense in question are precisely parallel to those which 
may be used against the word " direction " as applied to lines. I 
find, moreover, that Herschel, Grant, and other writers, use the 
word position as I have done, being apparently forced so to use it 


the moon travels were invariable, she would cross the 
ecliptic at the two fixed points which would be her 
nodes. During any single revolution of the moon 
this is not far from the actual case ; so that we may 
say without gross error that in a sidereal month the 
moon is twice on the ecliptic_, and twice at her greatest 
distance north and south of the ecliptic, that is, about 
5 8' (on the average) north and south of that circle. 
Viewing the matter in this way for the moment, let 
us inquire in what way the moon's range north and 
south of the equator, and her motions generally, as 
seen from the earth, are aflFected, according as her 
nodes lie in different parts of the ecliptic. 

Let SE NW (fig. 53, Plate XIII.) represent the 
plane of the horizon, N being the north point, and 
let S P N be the visible celestial sphere. Let 
E ^ W JSi be the celestial equator, the arrow on this 
circle showing the direction of the diurnal motion, 
and let WeEe' be the ecliptic, the arrow showing 
the direction of the sun's annual motion. The student 
will understand of course that the ecliptic is only 
placed, for convenience of drawing, in such a position 
as to cross the equator on the horizon at E and W. 
Twice in each day it occupies that position, as it is 

for want of any better word. Accordingly I retain the use of the 
word, and would suggest, as the best remedy against its defects, 
that writers should carefully avoid the use of the word to indicate 
'place, adopting instead the word situation. I give, then, this 
definition : Planes are said to have the same position when lines 
normal to them have the same direction. 

160 THE moon's changes 

carried round by the diurnal motion^ and once in each 
day it is in the exact position indicated in fig. 53 ; 
that is^ with its ascending node (or the first point ot 
Aries) just setting in the west. 

Now let us suppose that the rising node of the 
moon's orbit is at W, the place of the vernal equinox. 
Then W M E M' is the moon's orbit, e M and e W 
are arcs of about 5 9' ; and we see that the range of 
the moon north and south of the equator exceeds the 
range of the ecliptic (that is, of the sun) by these equal 
arcs. In other words, the moon when at M is about 
28 36' north of the equator instead of being only about 
23 2 7'north, as she would be if she moved on the ecliptic, 
while when at M' she is about 28 36' south of the 
equator : she moves throughout the sidereal month 
as the sun moves throughout the sidereal year, passing 
alternately north and south of the equator, but with 
a greater range, due to the greater inclination of her 
orbit. Accordingly, she remains a longer time above 
the horizon when at any given stage of the northern 
half of her orbit, and she remains a shorter time above 
the horizon when at any given stage of the southern 
half of her orbit than she would be if she moved on 
the ecliptic. She also passes higher than the sun 
above the horizon when at her greatest northerly 
range, attaining at this time (in our latitudes) a height 
of more than ^^, as at M, instead of less than 61 ; and 
she is correspondingly nearer the horizon in southing 
when at her greatest southerly range from the equator, 
attaining in fact a southerly elevation of less than 


If(js.55,66,^Z^nri58, illusfm^te the .Mooti's Axial R/)taUo/?. 


\h M 

1 p 








Ifhfshntin^ rJ?e Lihrrfwn of the &filrf; of fke .Moa/fls- /)/.sr . 

jR.A.Proclor ddS 


10^ as at m, instead of more tlaan 15, as is tlie 
case with the sun. 

Next let us suppose that the descending node of 
the moon's orbit is at W (fig. 53, Plate XIII.), the 
place of the vernal equinox; then Wm Em' is the 
moon's orbit ; e m and e m" are arcs of about 5 9' ; and 
we see that the range of the moon north and south of 
the ecliptic is less than the range of the sun by these 
equal arcs. Thus the moon when at m is about 
18 18' north of the equator instead of 23 27', and 
she is about 18 18' south of the equator when at m'. 
Thus she has a smaller range than the sun north and 
south of the equator. She never attains a greater 
elevation above the southern horizon than about 56 
as at m ; but, on the other hand, her least elevation 
when due south exceeds 20, as at fj, (the sun's greatest 
and least southing elevations, as at e and s, being re- 
spectively about 61 and about 15). 

Thirdly, let the rising node of the moon's orbit be 
near e, the place of the summer solstice (fig. 54, Plate 
XIII.) ; then e M <?' M' is the moon's orbit, which 
crosses the equator at two points, M and M', in 
advance of the equinoctial points Wand E.* We see 

* These points and the points w and m' are about 12^ degrees 
from the points E and W, being determined by the relation that 
they are points on the equator about 5 9' north of the ecliptic. 
If great nicety were required in the above explanation, we should 
have to take into account the fact that the moon's orbit has not 
exactly its mean inclination to the equator when the nodes are on 
the solsticial colure ; for the angle eMJEis not equal to the angle 
E ^E, the mean inclination in question. But considerations of 


162 THE moon's changes 

that its greatest range from the ediptic is attained 
nearly at the points e and e, and is therefore appre- 
ciably equal to the. sun's range. The circumstances 
cf the moon's motion must therefore resemble very 
closely those of the sun's, the chief difference result- 
ing from the fact that the nodes of the moon's orbit 
in the equator are some twelve or thirteen degrees 
in advance of the equinoctial points. 

Lastly, similar considerations apply when the de- 
scending node of the moon's orbit is near e, the 
moon's path being in this case em em', and its nodes 
on the equator some twelve or thirteen degrees behind 
the equinoctial points. 

Now let it be noticed that the moon's orbit passes 
through the complete cycle of changes (of which the 
above four cases are the quarter changes) in about 
18'6 years, the lunar node moving on the whole 
backwards on the ecliptic. Thus, if such a cycle of 
years begin with the moon's orbit in the position 
W M E M' (fig. 53, Plate XIII.), then in about a fourth 
of the cycle (that is, in about 4*65 years), the moon's 
orbit is in or near the position e'm' e m, fig. 54, the 
node having moved backwards from W to near e', or 
one quarter of a revolution. One fourth of the cycle 
later, that is, about 9'3 years from the beginning of 
the cycle, the moon's orbit is in or near the position 
Em'Wm, fig. 53, the node having moved still back- 
wards from e' to near E. Yet another fourth of the 

this kind nepd not detain us in a general explanation such as that 
ve are now upon. 


cycle later, or about 13 '95 years from its commence- 
ment, the moon^s orbit is in or near the position 
e M e'W , fig. 54, the rising node having shifted back- 
wards from E to near e ; and^ lastly, at the end of the 
complete cycle of 18'6 years^ the moon's orbit is in or 
near its original position. 

It is obvious that since, on the whole, the lunar 
nodes thus regrede, or, as it were, meet the advancing 
moon, she must cross her nodes at intervals some- 
what shorter than a sidereal month. In fact, sup- 
posing her to start from her rising node at the be- 
ginning of a sidereal month of 27*322 days, then at 
the end she has returned to the part of the ecliptic 
she had occupied at the beginning, while the node has 
regreded on the average by that amount which is due 
to a period of 27*322 days. This amount is easily 
calculated, since the node regredes through the 
complete circuit of the ecliptic in 6793*391 days : it 
is rather less than 1 27'. So that, estimating her 
motion with reference to her rising node, the moon 
completes a circuit and nearly a degree and a half 
over J in 27*322 days; hence she completes a nodal 
circuit in a period less than 27*322 in the proportion 
very nearly of 360 to 361^.* This period, called the 

* Or another and more exact way of viewing the matter is as 
follows : The moon advances at a mean rate of -^^ degrees per 
day, the node regredes at a mean rate of ^^^ degrees per day. 
Thus the diurnal advance of the moon with respect to the node is 
the sum of these two quantities, and we have only to calculate how 
often this sum is contained in 360 degrees to find the exact number 
M 2 

1()4 THE M001!f S CHANGES 

nodical montL^ amounts to 27'212 days. It follows 
that the mean interval between successive passages 
of the lunar nodes is about 13|- days. Accordingly, 
the moon must always be twice at a node in every 
lunar month of 29^ days, and may be three times at 
a node ; since, if she is at a node within the first 2"3 
days from new moon, she is again at a node within 
I5*9 days from new moon, and yet again within 
29*5 days, that is, before the next new moon. 

The effects of the eccentricity of the lunar orbit are 
too obvious to need any special discussion. The moon 
moves more quickly (in miles per hour) when in 
perigee than when in apogee, in the proportion of 
about 1 9 to 1 7 on the average ; but as she is nearer 
in the same degree when in perigee, her apparent 
rate of motion along her orbit is yet farther increased, 
and in the same degree, so that her motion in her 
orbit is greater when she is in perigee than when she 
is in apogee, in about the proportion of the square 
of 19 to the square of 17, or about as 5 to 4.^ (We 

of days in a mean nodical month. This number is obviously the 
reciprocal of ^^ + ^-1^. This method is clearly the correct 
method to pursue in all such cases. The rule may be thus ex- 
pressed : Let P, P' be the periods in which two objects which 
may be planets, nodes, perigee-points, and so on make a circuit 
of the same celestial circle, P' being greater than P : then the 

interval between successive conjunctions is the reciprocal of -73 p> 
if the objects move in the same direction, and the reciprocal of 
-,7 + p7 if they move in different directions. 

* When we wish to obtain a fair approach to the ratio of the 
squares of two large numbers differing by two, we ha ve a ready 


note, in passing, that 19 to 17 is about the ratio in 
which the moon's apparent linear dimensions are 
greater when she is in perigee than when she is in 
apogee, while 5 to 4 is the apparent ratio in which 
her disc when she is in perigee exceeds her disc when 
she is in apogee.) As the eccentricity of her orbit is 
variable, its mean value being about 0*055, while its 
greatest and least values are about 0"066 and 0*044, 
there is a difi'erent range in her rates of real and 
apparent motion, according to the amount of eccen- 
tricity when she is in perigee or apogee respectively. 
The actual maximum rate of the moon's motion is 
attained when she is in perigee and her eccentricity 
has its maximum value 0*066, while the actual mini- 
mum is attained when she is in apogee at such a time. 
The ratio between her real motions, under these 
circumstances, is that of 1,066 to 934, or about 8 
to 7 j the ratio between her apparent motions in her 
orbit being rather greater than 13 to 10. 

These variations are sufficiently great to modify, in 
a remarkable degree, the movements of the moon 
when considered with reference to the change f- om day 
to day in her apparent place in the heavens, and there- 
fore, in her apparent course from horizon to horizon. 
We saw that this must be so, when we inquired 

means in the following considerations : The ratio (a + 2)^ ; a" is 
nearly the same as the ratio (a -t- i) (a + 3) : {a 1) (a + 1) ; that 
is, is nearly the same as the ratio (a + 3) : (a 1). In the above 
case this gives 20 to 16, or 5 to 4. The real value of the ratio 
(17)2; (19)2 is not 4-fifths or -8, but -80056, which differs from 
8 by less than the fuurteen-hundredth part. 

166 THE moon's changes 

into the phenomenon called the harvest moon. It is 
manifest also that all the circumstances of eclipses, 
solar as well as lunar, must be importantly modified 
by the remarkable variations which take place in the 
moon's distance from the earth, and in her real and 
apparent motions. The eccentricity of the moon's 
orbit also produces very interesting effects in relation 
to her librations. If the perigee and apogee always 
held a fixed position with respect to the nodes of the 
lunar orbit, the peculiarities thus arising would be less 
remarkable ; but the continual shifting of the relative 
positions ofthenodesandapses (as theperigeeandapogee 
are called) causes a continual variation, as we shall see 
hereafter, in the circumstances of the lunar librations. 

Speaking generally it may be said that the lunar peri- 
gee advances at the mean rate mentioned in the pre- 
ceding chapter, that is in such a way as make a complete 
circuit in about 3232-575 days. Accordingly, applying 
considerations resembling those applied to her motion 
with respect to her nodes, we see that the period of 
her motion from perigee to perigee must exceed a 
sidereal month. Its actual length is found to be 
about 27*555 days. This is the mean anomalistic 
month ;* it exceeds the mean nodical month by 
rather more than the third part of a day ; or more 
exactly by 0"342 of a day. 

The actual motion of the perigee and apogee with 

* The actual interval betweejj the moon's passages of her perigee 
varies during the course of a year from about 25 days to about 
28^ days. 


respect to the nodes is very variable. As shown in the 
preceding chapter, the apses are sometimes advancing 
rapidly and they advance on thewholeorregredeonthe 
whole for several successive months while at others 
they are almost as rapidlyregreding, and the node itself, 
though on the whole regreding in every lunation, yet 
sometimes advances slowly for several successive days. 
Thus the perigee and rising node are sometimes moving 
the same way, at others in opposite ways ; they may 
be both advancing or both receding, or the perigee 
may be advancing and the apogee receding, or the 
perigee receding and the apogee advancing. We 
can see from figs. 42 and 45 (Plate XI.) how variable 
these relations are even when no account is taken of 
the advance and recession taking place during the 
course of individual lunations. However, so far as 
the mean advance of the perigee from the node is 
concerned, the case is sufficiently simple ; for the 
perigee advances so as to complete a revolution on 
the average in 3232"575 days, or 8*8505 years, while 
the node recedes so as to complete a revolution on 
the average in 18*5997 days. Thus the mean annual 
advance of the perigee is g^. of a revolution, while 
the mean annual regression of the perigee is ^j~ 
of a revolution. Adding these together we find the 
mean motion of the perigee w^ith respect to the node 
equal to g:^ of a revolution.* In other words, the 

* The agreement of the figures in the denominator of this fraction 
with the last four in the fraction representing the motion of the 
node is of course a merely accidental coincidence. 

168 THE moon's changes 

mean interval between successive conjunctions of the 
perigee and rising node is very nearly six years, 
falling short of six years in fact by but about three 
thousandths of a year, or almost exactly ] ^^ days.* 
The mean interval between successive conjunctions 
of the apses and nodes (without regard to the dis- 
tinction between apogee and perigee, rising node and 
descending node) is three years, wanting only about 
half a day, or more exactly wanting 13 h. 18*5 m. 

We are now in a position to discuss the effects of 
the moon's rotation. 

If the moon as she went round the earth turned 
several times round upon an axis nearly square to the 
level of her path, she would present every part of her 
surface several times successively towards the earth, 
precisely as the earth turns every part of her surface 
towards the sun in the course of a year. On the 
other hand, if the moon did not turn round at all 
as she went round the earth, we should see in turn 
every part of her surface, since at opposite sides 
of the earth she would necessarily present two 
opposite faces towards the earth. Since as a matter 
of fact it may be said (as a first rough account of the 
moon's appearance) that she turns always the same 
face towards the earth, it follows that she must turn 
once on an axis nearly square to the level of her path 
Rs she performs one complete circuit. 

* The mean interval between successive conjunctions of the 
perigee and the rising node is 2190'343 days, and in six years there 
are 2191*452 days ; so that the mean difference is I'lOO days. 


Thus let us suppose that the globe Mj (fig. 55, 
Plate XIII.) circuits round the globe E without 
any change of positio7i. Then when the moving 
globe has completed one-fourth of a revolution, 
A B D, and is at Mg, the points A, B, C, D 
will be in the position shown, B instead of A being 
towards E. When the moving globe is at M3, C will 
be towards E ; when the globe is at M4, D will be 
towards E ; and lastly, when a complete revolution 
lias been effected, A will again be turned towards E. 
Obviously, to keep A always directed towards E, the 
line Mg A should be shifted through a quarter-revo- 
lution to the position Mg B ; M3 A should be shifted 
through half a revolution to the position Mg C ; and 
M4 A through three-quarters of a revolution to the 
position M4 D, all these shiftings being made in 
the same direction, viz. in the direction A B D, 
which is the same as that in which the body itself is 

This is shown again in fig. 56, Plate XIIL, where 
we see that if the middle point of the disc of the 
moving globe is the same real point on this globe, as 
it travels through the positions Mj, Mg, M3, M^ .... to 
Mg, this globe must have turned in the manner shown 
in fig. 56 a, the radii in which to the points 1, 2, 3, 4, 
&c., are respectively parallel to the radii to M^, Mg, 
M3, M4, &c., all of which are directed upon the central 
orb E. 

But it may occur to some readers that although 
^undoubtedly if a globe were carried from the position 


Mj to Mg, fig. 55j and A C forcibly kept in the position 
indicated, there would be the change of face we have 
described, yet that in the nature of things if a body 
were set without rotation travelling round a central 
globe, it would as it went round turn itself a/.so, 
as if upon an axis, and so keep always the same 
face dii-ected towards the central globe. For ex- 
ample, if a rod extending from E and rigidly attaches 
to Ml, carried that globe round E in the manner 
indicated, then the face A would remain constantly 
turned towards E : may it not be, it might be asked, 
that as the globe moved under gravity round E the 
same thing would happen ? If the globe M^, initially 
at rest, were propelled by a blow directed exactly on 
the line B M^ with precisely the velocity corresponding 
to the circular orbit M1M2M3M4 under gravity, might 
not the result of the attractions exerted by E be to 
cause the globe Mj not only to go round E, but to 
turn itself always so as to have the same face directed 
towards E ? 

Now it is mathematically demonstrable that the 
attraction of E can have no effect whatever in causing 
the direction of the line A M to change as the body 
(supposed to be spherical ^) circles around E. But 
the considerations on which such a demonstration 
would be based are by no means so obvious as is com- 

* If the body be not spherical, forces tending to produce 'i 
rotation come into play ; but if the body has even only a roughly 
globular form, such forces are altogether too small to produce any 
appreciable amount of rotation during a single revolution. 


monly supposed. We shall not, therefore, present 
them here,* but proceed at once to mention two 
experimental proofs of the fact in question. The first 
experiment is very simple. Let a tolerably heavy 
ball be suspended by a long fine cord. Let it be left 
hanging until all signs of twisting have passed away ; 
then, having placed a mark upon the ball anywhere 
except near the top or bottom, cause it to swing in 
a circle, communicating this motion by means of the 
string held at a point high above the ball, so that 
no rotational movement can be imparted. It will be 
found that the mark continues always to be directed 
towards the same point of the compass, not turning so 
as always to bear in the same direction with respect to 
the centre of motion. The second was suggested by 
Galileo, who pointed out that if a body be set to float 
in a basin of water, and this basin be held out at arm's 
length while the holder turns round, it will be found 
that the floating body does not partake in the turning 
motion ; so that the side turned towards the holder 
of the basin at the beginning is turned directly away 
from him when he has made half a turn. It is, how- 
ever, by no means easy to carry out this experiment in 
a satisfactory manner, the most striking phenomenon 

* It may suffice to remark that if a body circuits round E in the 
manner shown in fig. 55, the total quantity of work done accords 
exactly with that due to the imparted velocity ; but if it moves in 
the manner shown in fig. 56, the amount of work done exceeds that 
due to the imparted velocity by the amount corresponding to one 
complete rotation of the body. 

172 THE moon's changes 

under ordinary conditions being the spilling of tHree- 
fourtlis of the water, or thereabouts. 

But a very effective experiment for those who feel 
doubts respecting the moon's rotation may be con- 
ducted as follows : Let AB (fig. 57, Plate XIII.) be 
a flat wooden bar of any convenient dimensions (accord- 
ing to the circumstances under which the experiment 
is to be conducted). Let fig. 58 present a side view 
of the same bar, which, it will be observed, is arranged 
to run on casters ab A and B, and to turn on a pivot 
at C. At A let a small circle and arrow be marked 
on the bar ; at C and B let small basins of water be 
placed, in which let small wooden rods float, or pre- 
ferably, let the rods float in half-filled saucers, them- 
selves floating in the basins. If now the experimenter 
wait until the water is still, the floating rods being- 
central and parallel to the arrow at A, and if he then 
gently turn the wooden bar round on its pivot at C, 
the casters rolling on a smooth table or floor, he will 
see that the rods floating at B and C both retain a 
direction almost wholly unchanged throughout the 
motion ; and thus while continuing parallel to each 
other and also to any line on the table or floor to which 
they were parallel in the first instance, they no longer 
continue parallel to the arrow at A, whose direction 
changes throughout the motion. The slight change 
of position they undergo is obviously referable to 
friction between the water and the basins and saucers. 
Of course the basin C is not essential in this experi- 
ment, nor the fixed arrow at A. If the basin B were 


simply carried round the end A as a centre, a similar 
result would follow. But it is interesting to show that 
so far as the rotation of the water within the basin is 
concerned, the condition of the basin B is exactly the 
same as that of a basin at C turning simply on a 
pivot immediately beneath it. 

Another experiment may be tried with the same 
apparatus. The water in C and B may, without much 
trouble, be set rotating at the same rate. If this be 
done, and then the rod be carried round at the same 
rate, so that the floating rod in C retains an un- 
changed position with respect to the bar A B or to 
the arrow at A, it will be found that the water in B 
behaves precisely as the moon\s globe behaves (so far 
at least as the general relation we are dealing with is 
concerned), turning always the same portion towards 
the centre C. Thus we learn that it is only by an 
additional rotational movement that such a relation 
can be preserved. 

The moon then turns once upon her axis as she 
completes the circuit of her orbit. Yet it is not 
strictly the case that the moon turns always the same 
face towards the earth. We see somewhat more than 
one half of the moon's surface. Let us inquire how 
this is brought about. 

In the first place, the moon's axis is not at right 
angles to the plane of the path in which she travels 
round the earth. (Let it be noticed, in passing, that 
it is the inclination of the moon's axis to this plane, 
and not to the plane of the ecliptic, which affects her 


appearance as seen from tlie earth. This will appear 
obvious as we proceed.) 

The moon's equator-plane is inclined 1 30' IT' to 
the plane of the ecliptic, and is always so placed that 
when the moon is at the ascending or descending 
node of her orbit, the equator-plane is turned edge- 
wise towards the earth, and is inclined descendingly 
or ascendingly (respectively) to the ecliptic. In other 
words, if eM.e (fig. 61, Plate XIV.) represent the 
ecliptic, M being the rising node of the moon's orbit, 
MM', then the moon's equator is in the position 
E E'; while, if M is the descending node (fig. 63), then 
the equator is in the position E E' (fig. 63); the 
angle e M E in both cases being one of 1 30' 1 1". 
Since the average inclination of the moon's orbit to 
the ecliptic is nearly 5 9', it follows that the angle 
EMM' has a mean value of about 6 39'; but this 
angle varies as the inclination of the moon's orbit 
varies, and is sometimes as great as 6 44',* sometimes 
as small as 6 34'. 

Now the effect of this inclination of the moon's axis 
to the plane of her orbit about the earth corresponds 
precisely to the seasonal variations of the earth's pre- 
sentation towards the sun. Thus we see that as the 
moon passes away from the position shown in fig. 61, 

* I find commonly 6 47' set as the value of this angle. This 
seems to be obtained by adding the moon's maximum orbit incli- 
nation 5 17' to the inclination of her axis to the ecliptic. But the 
moon is always near a node when her orbit attains its maximum 
inclination, whereas the maximum opening due to her inclination is 
attained when she is farthest from her nodes. 



Plate XIV., moving towards the left, the pole P will 
be brought into view, and the moon's equator will 
open out with its convexity downwards, so that at the 
end of a quarter of a revolution (from rising node to 
rising node) the aspect of the moon will be as shown 
in fig. 62. At the end of another quarter of a revo- 
lution, when the moon will again be at a node, her 
aspect will be as in fig. 63 ; at the end of the 
third quarter as at fig. 64 ; and when the revolution 
is completed she will again be as at fig. 61. We see, 
then, that her face varies on account of her inclination, 
the middle of her visible disc Ijnng about 6 39' north of 
the equator, when she presents the aspect shown in 
fig. 62, and as far south of the equator when she presents 
the aspect shown in fig. 64. Here no account is taken 
of rotation, precisely as in dealing with the terrestrial 
seasons we consider separately the seasonal changes 
of the earth's aspect and the effects of her rotation. 

Seeing that the middle of the disc passes alternately 
north and south of the moon's equator, or, which is 
the same thing, that M, the middle point of the visible 
half of the equator, passes alternately south and north 
of the centre of the disc, we are led to inquire at 
what rate this oscillatory motion takes place at different 
parts of the nodical month. The mathematician 
will find no difficulty in proving the following re- 
lation : * 

* See note on pp. 80 and 81 of my treatise on Saturn for consi- 
derations rendering the solution of all such problems exceedingly 

176 THE moon's changes 

Let Mj, fig. 59, Plate XIII., represent the middle of 
the disc when the moon shows the aspect indicated in 
fig.^. 61 and 63, Plate XI Y. ; M^ the middle of the 
visible half of the equator under the aspect shown in 
fig. 62 ; and Mg the same point under the aspect shown 
in fig. 64. Then if a circle, A Mg B M3, be described 
about Ml as centre, and a point be supposed to 
traverse this circle at a uniform rate, in a nodical 
month, starting from A when the moon is at a rising 
node ; and if P be the position of this point at any 
part of such a month, then P M drawn perpendicular to 
M2 Mg gives M, the position occupied by the middle 
of the visible half of the moon's equator at that 
moment. Thus we see that the middle point of the 
moon's equator oscillates northwards and southwards 
(along the apparent projections of the moon's polar 
axis), moving very slowly near M2 and Mg, and most 
quickly in crossing the point Mj, the middle of the 
moon's disc. 

It will be easily seen that, considering only the 
efiects due to the moon's inclination, fig. 69, Plate XY., 
represents the changes affecting points which, in the 
moon's mean state (as in figs. 61 and 63, Plate XIY.), 
occupy the middle of the short vertical lines of that 
figure. Supposing the nodical month divided into " 
twelve equal parts (each, therefore, about 2^ days in 
length), these points would travel upwards and down- 
wards along the vertical lines of fig. 69 (which lines 
are, of course, not really vertical on the moon's disc 
as we see it), traversing the points there marked in 



Pi/. 7(1 

E mrr 


Jlhoitra/in/f M^p' 

Lunar L ib ration. 

t E 

of ifariouipomts onlhe ttwoji, 
C D t r G 

|^ \ .cerdretfdisc 

^ 4V 






























e'^- X. 





point , 




the order indicated by tlie numbers on tbe central 
short line. 

But now let us take into consideration the effect of 
the want of perfect accordance between the moon's 
motions of revolution and rotation. She rotates uni- 
formly on her axis, or very nearly so, while she moves 
with varying velocity round the earth. But fig. 56, 
Plate XIII., shows, that in order that the same face 
should always be shown, there should be perfect 
agreement between the motions of rotation and 

Let M1M2M3M4 (fig. 70, Plate XV.) be the lunar 
orbit about E, the earth, M^ being the perigee of 
the orbit and C its centre, the shape of the orbit (see 
note, p. 27) being appreciably circular. Then the 
moon moves more rapidly over the arc Mj B than over 
the arc B M3 ; and therefore, at the end of a quarter 
of a revolution she is not at B, but in advance of that 
point. To find how much, we have only to consider 
that in a quarter of a revolution a line from the earth 
to the moon sweeps over a quarter of the area 
M1M2M3M4. Hence, if M.^ be the moon's place at 
the end of a quarter of a revolution, the area Mg E Mj 
is equal to the quarter B C Mj of the complete orbit. 
So that the triangle ODE must be equal to the space 
B D Mg. This will obviously be very nearly the case if 
D is the middle point of the line C B (see further the 
note on p. 107). And in like manner, we obtain the 
point M4, reached by the moon after three-quarters of 
a revolution, by drawing ED'M^ through the middle 



point D' of E B'. Now, it will be readily conceived 
that since tlie moon when at B is at her mean distance, 
she is travelling nearly at her mean rate in the 
neighbourhood of this point (her orbit being nearly 
circular in shape), so that at Mg she is no longer 
getting in advance of her mean place, and has, there- 
fore, attained her maximum displacement in advance. 
In like manner, when she is at M4, she has attained 
(approximately) her maximum displacement behind 
her mean place. And it is very easy to find the 
effects (necessarily maximum effects, at the points 
M2 and M4) of the non-accordance between the motions 
of rotation and revolution. If the moon swept at a 
uniform rate round the point E, she would be at P and 
P' at the times when, in reality, she is at M2 and M4 
(P E P' being drawn at right angles to C E). This is 
obvious, since the four angles P E M^, P E M3, FE M3, 
and P'EMi are all equal, and the moon occupies 
equal times in going from Mi to M^, thence to M3, 
thence to M4, and thence, finally, to Mj again.* So 

* In fact, we are assuming that P E Mj, P'E M4, represent the 
maximum values of the difference between the true and the mean 
anomaly, or, with ordinary notation, the maxinmm values of {9n t). 
Now it is obvious that the circular measure of the angle P E Ma is 
very nearly represented by -^^, or by 2 e. Hence we are assuming 
that the maximum value of - w t is very nearly equal to 2 e. In 
reality, this value is represented by an infinite series, beginning 


For the mean value of the lunar eccentricity, the term involving e 
amounts only to 0-00003792, or less than the 2,895th part of 2 e. 


that, if we draw magnified pictures of the moon at 
Mj, M2, M3, and M4, and put m^ as the point nearest 
to the earth (or the middle of the moon's visible disc) 
when the moon is at Mj, the line Mj m^ will have 
shifted so as to be parallel to P E when the moon is 
at Mjj, in other words, it will have the position 
M2 W2> ^^^^ instead of being the middle of the visible 
lunar disc, Wg will be displaced in the direction in 
.which the moon is moving, or towards the east. 
When the moon is at Mg, the same line will have made 
half a turn, or be in the position M3 m^ ; directed 
thereforOj as at first, towards the earth. When the 
moon is at M4, the same line will be parallel to P'E, 
or have the position M4 m^, and instead of being the 
middle of the moon's visible disc, m^ will be displaced 
towards the west. When the eccentricity of the 
moon's orbit has its mean value, the value of the 
angle PE Mg or FE M^ is about 6 17' 19"-04. But 
when the eccentricity has its maximum value, the 
angle PEMg or P'EM4 amounts to 7 20', and owing 
to lunar perturbations, it may be increased to so much 
as 7 45'.* 

It follows then that the effect of the want of accord- 
ance between the moon's rotation and revolution is to 
sway the lunar meridian through the middle point of 

* This is the result of my own calculations. I find 7 53' and 
7 55' set by different authorities as the greatest value of the angle 
in question. It appears to me that the circumstance has been over- 
looked that the moon's orbit never has its maximum eccentricity 
when the moon is at her mean distance. 
N 2 

180 THE moon's changes 

the disCj when the moon is in perigee or apogee, in the 
manner indicated in figs. 05, 66, 67, and 68. Here no 
account is taken of the change of aspect due to the 
moon's inclination, but the polar axis is supposed to be 
throughout in an unchanged position. When the moon 
is in perigee, this meridian has the position shown 
in fig. 65, Plate XIV. A quarter of a revolution after 
perigee it has the position shown at fig. 66 on the east 
of the mean position. When the moon is in apogee, 
the meridian is again central, or as in fig. 67 ; a 
quarter of a revolution later, it has the position shown 
at fig. 68, to the west of the mean position ; and 
lastly, when she is again at her perigee, the meridian 
is again as shown at fig. 65. 

In this case, also, seeing that the point M passes 
alternately east and west of the middle of the disc, we 
are led to inquire at what rate this oscillatory motion 
takes place in different parts of the anomalistic 
month. The mathematician will find no difficulty in 
proving that, approximately, the law of this oscillatory 
motion is similar to that of the libration due to the 
moon's inclination : * 

* If M (fig. 70, Plate XV.) be the position of the moon in any part 
of her orbit, it is easily shown that C dp, having d as its middle 
point, shows the direction (approximately) in which the moon would 
have been at the moment if she had circled at a uniform rate 
around C. Thus, whereas C D E (measured by E C) gives the 
maximum libration, C d E (measured on the same scale by E k) 
gives the libration when the moon is at m. Now the ratio E k 
by E C is the sine of the angle p C Mi, which proves the above 


Thus, let Ml fig. 60, Plate XIII., represent the middle 
of the disc when the moon is in perigee or apogee, or as 
shown in figs. 65 and 67, Plate XIV. ; Mg, the position 
of the same point (corresponding to M) when the moon 
is as shown in fig. 6Q ; and Mg, the position of the same 
point when the moon is as shown in fig. Q8. Then 
if a circle A Mg B Mg be described about M^ as centre, 
and a point be supposed to traverse this circle at a 
. uniform rate in an anomalistic month, starting from A 
when the moon is in perigee ; and if P be the posi- 
tion of this point at any part of such a month, P M 
drawn perpendicular to Mg Mg gives M the position 
occupied at that moment by the point which had been 
at the middle of the visible disc when the moon was 
in perigee. Thus, this point oscillates eastwards and 
westwards (in a direction at right angles to tho 
apparent projection of the moon's polar axis), moving 
very slowly near Mg and Mg, and most quickly in 
crossing the point M^, the middle of the moon's disc. 

It will easily be seen that considering only the 
effects due to the moon's variable motion in her 
orbit, fig. 71, Plate XV., represents the changes 
affecting points which, in the moon's mean state, 
occupy the middle of the short horizontal lines of the 
figure. Supposing the anomalistic month divided into 
twelve equal parts (each therefore about 2 5 days in 
length), these points would travel forwards and back- 
wards along the horizontal lines of fig. 71 (which 
lines are of course not really horizontal on the moon's 
disc as we see it), traversing the points thus marked 

182 THE moon's changes 

in the order indicated by tlie numbers on tbe central 
short line. 

It remains for us to determine the combined effects 
of the movements here separately dealt with. Fully 
to treat the matter in this general aspect would 
require much more space than can here be given ; 
moreover, the problems involved are not quite suit- 
able for these pages. But a suflBcient idea of the 
general effects of libration can be obtained by at- 
tending to the following considerations : 

In the first place, let it be noticed that we need not 
concern ourselves at all about the varying slope of 
P Vj as illustrated in figs. 61, 62, &c., but need attend 
only to the consideration that the moon librates, owing 
to her inclination, precisely as though swaying on an 
axis through the points E E' on the edge of her disc. 
In like manner, owing to her varying rate of rotation, 
she librates precisely as though swaying on an axis 
through the points P P', on the edge of her disc. 

In the next place, let it be noticed that everywhere 
over the moon's disc, except at points so near to the 
edge that the libration actually carries them at times 
out of sight, the effects of the two forms of libration 
can be obtained by combining the two figs. 69 and 71, 
as in fig. 72, and noting that the small crosses indi- 
cate the double oscillation of the several points at the 
intersection of the cross-lines, and that such double 
oscillation is combined into a single oscillation, whose 
nature at any instant depends on the relation existing 
at the moment between the moon's motion from 


rising node to rising node, and from perigee to 
perigee. If we consider these effects for the middle 
point of the disc, we shall be able to infer their 
nature for points anywhere on the disc. 

Let AGG'A', fig. 73, Plate XV,, represent a 
magnified view of the small space at the centre of 
PEP'B. Divide A A' in the manner indicated (de- 
scribing a semicircle on this line and dividing the 
semicircle into say six equal parts, drawing per- 
pendiculars on A A' from the points of division) and 
A Gr similarly. This division corresponds to that 
illustrated by figs. 69, 70, and 71. Draw parallels 
through the division- points so as to make the com- 
plete series of rectangular divisions shown in fig. 73. 

Suppose that the moon is at a rising node and 
also in perigee, so that there is no libration either in 
longitude or latitude. Then the centre of the moon^s 
disc is the true centre of the portion of the moon's 
surface discernible from the earth. This point is 
at 0, the centre of the rectangular space A Gr Gr' A'. 

Now as the moon advances on her orbit this central 
point (which for convenience we may call the mean 
centre) is carried eastwards of O, because the moon 
has just passed her perigee, and southwards of O 
because she has just passed her rising node. The 
first motion would carry it to K, L, d, at intervals 
each equal to a twelfth part of the anomalistic month ; 
the second would carry it to m, 7^, D', at intervals 
each equal to a twelfth part of a nodical month. 
Assuming for the moment that these months are 

184 THE moon's changes 

equal, whicb we may do without important error so 
far as a single revolution of the moon is concerned, 
we see that the moon will be carried along the line 
O A', reaching the points or stages indicated along 
that line at intervals each equal to about 2j days. It 
will be carried by the continuance of the same com- 
bined librations back to 0, which it will reach when 
the moon is in apogee and at her descending node, or 
half a mouth (mean nodical and anomalistic) from 
the beginning of the motion ; then it will pass north- 
wards and westwards to G, and so back to at 
the end of the month. 

Thus, in the imagined state of things, the mean 
centre sways backwards and forwards along the line 

Now, as a matter of fact, the mean nodical month is 
shorter than the mean anomalistic month. Therefore 
the moon, starting under the conditions just de- 
scribed, will presently so move as to reach her rising 
node some time before she reaches her perigee. Let 
us suppose the node to have separated 30 (a twelfth 
part of a complete circuit) from the perigee, which 
will happen on the average almost exactly half a year 
from the time when these points coincided. 

Now, therefore, the moon comes to her rising node 
when the mean centre is still west of the centre 
of the disc by the amount due to one-twelfth of the 
anomalistic month ; whereas, when she is at her 
perigee, the mean centre is south of the centre of the 
disc by the amount due to a twelfth of a nodical 


month. Accordingly, when she is at the rising node, 
the mean centre is at M, and when she is at her 
perigee the mean centre is at m. And it is very easy 
to see that, supposing the node and perigee to retain 
this position throughout the month, the mean centre 
traverses the oval M B'K V in that direction. 

By like reasoning it is obvious that when the rising 
node is 60 behind the perigee, the mean centre 
traverses the oval N C L c' in that direction. 

When the rising node is 90 behind the perigee, 
the mean centre traverses the oval (VD'dJ) in that 
direction. At this time there is no libration in longi- 
tude when the libration in latitude is at a maximum, 
and no libration in latitude when the libration in 
longitude is at a maximum. On the average, almost 
exactly a year and a half has now passed from the 
time when the mean centre librated along the line 

"When the rising node has regreded 120 from the 
perigee, it is clear that when the moon is at a node 
the westerly libration has not reached its maximum. 
The mean centre is then at N, and moving south- 
wards and westwards. It traverses then the oval 
Ne'E'LcC in that direction. 

When the risino^ node has reg-reded 150 from the 
perigee, the mean centre traverses the oval M/'K?> 
in that direction. 

When the rising node has regreded 180 from the 
perigee, or coincides with the apogee, the mean 
centre again lib rates linearly over O, but on the line 

186 THE moon's changes 

A G\ This happens almost exactly three years on 
the average from the time when the rising node and 
perigee were in codj unction. 

Still regreding, the node passes 30 behind the 
apogee, at which time the mean centre traverses 
the oval KF'MB in that direction. It will be no- 
ticed that throughout the former series of changes, 
the direction of its motion around was the same as 
that of the hands of a watch. Now the direction 
is reversed, and continues so during the series of 
changes taking place as the rising node passes from 
coincidence with the apogee to coincidence with the 

When the rising node is 60 behind the apogee, 
the mean centre traverses the oval L E'N C in that 

When the rising node is 90 behind the apogee, 
the mean centre traverses the oval dD'd'D in that 
direction. At this time the state of things corre- 
sponds with that which prevailed when the rising 
node was 90 behind the perigee, except that the oval 
having axes D D and d d^ is traversed in the opposite 
direction. An interval of almost exactly 4^ years has 
(on the average) now passed since the rising node and 
perigee were in conjunction. 

When the rising node is 120 behind the apogee, 
the mean centre traverses the oval LC'NE in that 

When the rising node is 150 behind the apogee. 


the mean centre traverses the oval KB'M6'in that 

And lastly^ when the rising node is again coincident 
with the perigee, the mean centre moves backwards 
and forwards along the line A'OGr, as at the begin- 
ning of the period. This period is on the average 
almost exactly six years. 

It will be easily seen how changes corresponding 
with those just described take place for every point 
on the moon's disc. If we call the intersection of 
any of the small cross-lines in fig. 72, Plate XY., 
a mean point, this mean point sways over and round 
its mean position precisely as the mean centre sways 
over and round 0, only that the ovals described 
differ in shape from A G Gr' A, and are also less 
symmetrical in figure. Fig. 74, Plate XY., illustrates 
the motions for a point on P P', and not far from P ; 
fig. 75, for a point on E E', and not far from E ; while 
fig. 76 illustrates the motions for a point not far from 
the point Mg. It will be understood that the letters 
in all these figures correspond with those in fig. 73 ; 
so that when the mean centre is at A,D, or G (Hg. 73) 
for example, the mean points corresponding to 
figs. 74, 75, and 76, are at A, D, or G, on those several 
figures respectively. 

Now since, as a matter of fact, the rising node does 
not move from the perigee by sudden shiftings of 30, 
it follows that the path traversed by the mean centre 
shifts gradually from one oval to another of fig. 73 ; 

188 THE moon's changes 

not completing any one of these ovals, but so moving 
tliat one oval merges into the next in a continuous 
manner. But since the node is not al\va3^s regreding 
nor the perigee always advancing, there is not a 
steady advance from one shape of loop to the next, 
but an alternation of advance and retrogression as 
respects the completion of the series of changes. In 
traversing a single circuit, indeed, there is always a 
double alternation, and sometimes a more complex 
series of alternations of this sort; because the perigee 
alternately advances and regredes twice in each lunar 
revolution, the node doing likewise, though in a less 
marked degree. But these effects are insignificant 
compared with those due to the regression of the peri- 
gee for several successive months, as explained in the 
preceding chapter, and illustrated by fig. 42, Plate .XI. 
This causes for the time being a reversal of the 
effects we have been considering; so that we have in 
every interval between successive conjunctions of the 
perigee and sun (or in every period of 411 days) two 
periods when the processes of change in the loops of 
fig. 73, Plate XY. (as well of course as in those of 
figs. 74, 75, and 76) are reversed for the time being. 
Adding to this consideration the circumstance that 
the eccentricity and inclination both undergo alter- 
ations (so that the length and bieadth of A G G'A' 
are variable), and that nearly 80^ nodical months 
occur between successive conjunctions of rising node 
and peri<;ee, we see that the path actually traced out 
by the mean centre is exceedingly complicated. The 


motion of this poi7it involves implicitly the whole theory 
of the muon^s motions. 

We have not considered tlius far the effects of 
libration on those parts of the moon's disc which lie 
so near to the edge that they pass at times out of 
sight. These movements might be dealt with like 
those we have just been considering, by regarding 
them as due to two distinct libratory motions taking 
place about P P' and E E' at known rates. But the 
matter may be simplified by noting that where (as in 
the present instance) such small arcs as from 6 to 10 
are CDncerned, the libratory motions of points near the 
rim of the mean lunar disc may be regarded as vir- 
tually carrying those points backwards and forwards 
at right angles to the rim. And it is very easy to see 
what will be the extent of this libratory motion at any 
given part of the rim when the libration of the mean 
centre is known. Thus, take the point P (fig. 72, 
Plate XY.). Here there is always an alternate sway- 
ing at right angles to the rim equal in range to the 
libration in latitude ; for whatever the oval traversed 
by the mean centre, it always ranges in latitude from 
A G to A' G'. In like manner at E and E' there is 
always an alternate swaying at right angles to the 
limb equal in range to the libration in longitude ; for 
the cval traversed by the mean centre always ranges 
in longitude from A A' to G G'. But take the part 
Mg of the disc's edge. Here, when the mean centre 
is librating along A G', points near the edge sway 
backwards and forwards at right angles to the edge 


over a range equal to tlie maximum libratory swing 
A G' (foresliortened of course, so as never to bring 
sucli points far within the edge in appearance). But 
when the mean centre is librating along A'O G, points 
near M are scarcely shifted at all. In intermediate 
cases, points near Mg have an intermediate range. 
Thus when the mean centre is traversing the oval 
M F K B', the range of points near Mg is equal to the 
breadth of this oval measured parallel to Mg M^. 
These remarks apply unchanged to points near M. 
At points near M^ and Mg corresponding changes 
take place ; only it is when the mean centre is libra- 
ting along A'O G that points near M^ and Mg sway 
over the largest arc across the rim of the disc, and 
when the mean centre is librating along AOGr' that 
these points remain nearly at rest. No point has any 
libratory motion along the rim of the disc.^ 

Such are the chief features of the lunar libra- 

* By the principles of rotation, we know that since under all 
circumstances the libratious in latitude and longitude take place 
about the axes E E' and P P', the actual libratory motion at any 
moment must always be about an axis in the plane PEP'E'. 
And it is very easy to determine the momentary position of that 
axis, as well as the actual circumstances of the displacement of the 
moon from its mean position. Thus let 0, fig. 77, Plate XV., be 
the centre of the moon's disc, O' the position of the mean centre at 
the moment, 0' T a tangent to the direction in which the mean 
centre is at the moment moving. Then K L at right angles to 
P T is the momentary axis of rotation, and the actual displacement 
of points on the moon's surface at the moment is the same as would 
have been produced by rotating the moon from its mean position 
about an axis M N at right angles to 0', so that the mean centre 


tions in latitude and longitude. It remains that 
we should consider what is the actual extent of the 
moon's surface which these librations bring into view 
in addition to that which is seen when the mean 
centre is at the actual centre of the lunar disc. In 
making the inquiry, we must take into account another 
libration, called the diurnal libration, which depends 
on the circumstance, that owing to the earth^s rotation, 
the place of the observer is shifted with respect to 
the line joining the centres of the earth and moon. 
This form of libration might very well be made the 
subject of a separate investigation, which would, how- 
ever, be more tedious than profitable, because the 
extent and nature of the diurnal libration varies in 
different latitudes and at different seasons. On this 
point, I shall content myself with remarking that if 
we imagine an observer placed at the centre of the 
moon's visible disc, a line drawn from him to any 
station on the earth would be carried by the earth's 
rotation along a latitude-parallel, and the angle which 
it made at any moment with a Hue joining the centres 
of the earth and moon would correspond to the 

was carried from to 0'. Thus points M and N are at their mean 
place, and points F and G are shifted by an arc equal to 0' from 
their mean position. The point which in its mean position would 
be at F is behind the disc, and the point which in its mean position 
would be at G has advanced on the disc as to g {G g being an 
arc equal to 0', but foreshortened). It is obvious that wherever 
0' may be (except at 0), two points only, and those both on the 
edge of the disc, are in their mean positions (as M and N in the 
case illustrated by fig. 77). 


diurnal displacement of the moon's centre, as seen 
from the station at that moment. This consideration, 
combined with what will hereafter be stated respecting 
the aspect of the earth as seen from the moon, will 
suffice to show the exact nature of the diurnal libra- 
tion at any given station, and at any season. Here, 
however, all that is necessary to be noticed is that, 
since the earth's radius, as supposed to be seen from 
the moon, subtends nearly a degree when the moon is 
at her mean distance, and more than a degree when 
the moon is in perigee, we may obviously add an arc 
of about a degree on the moon's surface to any libra- 
tory displacement in any direction whatever, estimated 
for the centre of the earth, if we wish to determine 
the maximum displacement in that direction /o?' any 
part of the earth. For, if we suppose an observer on 
the moon to shift his place, in any direction, by one 
lunar degree (corresponding to a distance of nearly 
twenty miles), he would see the earth's centre shifted 
one degree on the heavens ; and, therefore, the point on 
the heavens formerly occupied by the earth's centre 
would now be occupied by a point on or very close 
to the circumference of the earth's disc. Therefore, 
when we have determined the fringe of extra surface 
brought into view by the moon's maximum librations, 
we can widen this fringe all round by a breadth of 
about one degree. We must not indeed widen it 
everywhere by a breadth of 1 1' 24^', the maximum 
apparent semi-diameter of the earth as seen from the 
moon, simply because this apparent semi-diameter ia 


only presented wlien the moon is in perigee, while the 
moon attains her greatest total libration (corresponding 
to the displacement of the mean centre from to A, 
or A', or G, or G', fig. 73), as well as her greatest 
libration in longitude (corresponding to the displace- 
ment of the mean centre from to D, or d, or D', or 
d', fig. 73), only when she is at her mean distance. 
We may, however, employ even this maximum value 
of the horizontal parallax when the moon has her 
maximum libration in latitude, since there is nothing 
to prevent her from attaining this libration when she 
is at her nearest to the earth. These considerations, 
however, are unimportant, compared with those de- 
pending on the moon^s librations in longitude and 
latitude, simply because the diurnal libration or, as it 
may more fitly be termed, the parallactic libration 
attains its maximum only when the moon is on the 
horizon, and therefore very ill-placed for telescopic 

In considering the actual extent of the moon's 
surface, which her librations carry into and out of 
view alternately, we need not trouble ourselves about 
the varying nature of the combined libration. It 
might seem, at first sight, as though certain parts of 
the moon would only be brought into view while the 
libration in latitude attains its maximum value, that 
is, when the libration in longitude vanishes; and vice 
vco'sd. But as a matter of fact, if we consider the four 
cases where the total libration has its absolute maxi- 
mum value viz., when the mean centre is at the four 


points A, Gr, A', and G' (fig. 73, Plate XV.), we take 
into account every portion of the moon^s surface 
whicli libration can possibly bring into view. 

Thus in fig. 78, Plate XYI., PEFE' represents 
the outline of the moon^s disc, A Gr A' G' the space 
over which the mean centre shifts owing to libra- 
tions. Assume that when the mean centre is at 
A, G, A', and G', successively, the lune-shaped spaces 
brought into view are represented by the four cres- 
cents a Mg g mg, a Mj g' m^, a Mg g' m^, and a M4 g m^ 
(in reality, on the globe itself, these lunes have at the 
points Ml, P, Mg, &c., the breadth indicated in fig. 78). 
Now it is easy to determine the breadth of these 
lunes at any distance from the points Mi, Mg, Mg, and 
M4, where they are widest. Thus we know that the 
ratio of P p to Mi mj is the cosine of the angle 
P Mj. But Ml mi is equal to G, and the ratio of 
O D to G is as the cosine of the same .angle 
POMp* Hence Pp is equal to OD that is, we 
get the same libratory displacement (on the sphere, of 
course) of the point P, by taking the maximum libra- 
tion of the mean centre (either to G or A) as though 
we took the greatest libration in latitude, or D 
alone. And similarly with the greatest libration in 

* If we circumscribe the figure D cl, as shown in fig. 78, it can 
readily be seen that any great circle taken as D P p is taken, 
will have the portion within the small circle corresponding to the 
portion D equal to the portion within the lune a mi g' Mi (cor- 
responding to P p). A similar remark applies to circles circum- 
scribing the figures D d, d D', and D' d\ 


FuL 78. 


p' p' f*' 


longitude : it cannot shift E or E' more than they are 
shifted when is carried either to A or A' on one 
sidcj or to Gr and G' on the other. 

So that all we have to ascertain is the area of the 
space on the sphere corresponding to that^ in fig. 78, 
between the circle P E P'E' and curves p e, e p', p' e', 
and e'p. This is easily effected,* and we learn that 

* We know that the maximum breadth of the four lunes viz., 
the breadth at Mi, Mo, M3, and M4 is 10 16'. So the area of 
each opposite lune bears to the area of the whole sphere the ratio 
which 10 16' bears to 360". Now, by a well-known property of 
the sphere, the space PpMimi bears to the half-lune amiMj a 
ratio equal to the sine of the angle P Mj, equal therefore to G D 
by G. But G D represents an arc of 7 45' on the sphere, while 
O G represents an arc of 10 16'. Thus the area P mj bears to the^ 
whole sphere the ratio which 7 45' bears to twice 360. In like 
manner, the area E' mi bears to the whole sphere the ratio which 
6 44' bears to twice 360. This gives us the area of the space 
P p E' e', which is one-fourth of the total area brought into view by 
libration. Thus this total area bears to the whole sphere the ratio^ 

2 (7 45' + 6 44') : 360 
= 14 29' : 180 
= 869 : 10800 
= 1IT35 or -08046 

The total area brought into view by libration bears to the hemi- 
sphere invisible at the time of mean libration the ratio of about 
100 to 621. (Arago makes the ratio 1 to 7, though using 10 24' as 
the absolute maximum of hbration.) It is not easy to understand 
how an error crept into his treatment of a problem so simple. 
The proportion of the part of this hemisphere never seen to the 
whole hemisphere is therefore about 521 to 621 ; or if we represent 
the whole sphere by 1, the area of the part absolutely invisible will be 
represented by '4198. (Klein, in his " Sonnensystera," gives '4243, 
which is nearer to the truth than the value resulting from Arago's 



the area thus brought into view by libration is 
between one-twelfth and one-thirteenth of the whole 
area of the moon, or nearly one-sixth part of the 
hemisphere turned away from the earth when the 
moon is at her state of mean libration. Of course 
a precisely equal portion of the hemisphere turned 
towards us during mean libration is carried out of 
view by the lunar librations. 

If we add to each of these areas a fringe about 
1 wide,, due to the diurnal libration, a fringe which 
we may call the parallactic fringe, since it is brought 
into view through the same cause which produces the 
lunar parallax, we shall find that the total brought 
into view is almost exactly one-eleventh part of the 
whole surface of the moon ; a similar area is carried 
out of view : so that the whole region thus swayed 

estimate, namely, "4286, yet still considerably in error, particularly 
as Klein also names the value 10 24' for the maximum libration.) 

If, however, we take into account the effects of the diurnal libra- 
tion, it can readily be shown that the portion of the moon which is 
never seen under any circumstances bears to the area of the whole 
moon almost exactly the proportion which 148 bears to 360, or 37 to 
90, that is, it is equal to 0*4111 of the whole area. The part which 
can be carried out of view or into view by the libration, including the 

Q 1 

parallactic libration, amounts to ths of the whole surface, or p- 

if the whole area is represented by unity. 

The above numerical results have been carefully tested, and can 
be relied on as strictly accurate. It is easy for the reader to re- 
examine them. It may be noted, that instead of the above method 
for determining the area brought into view by libration we may 
simply add to the two lunes a Mj fj' and a M3 g', the spherical 
triangles e ma P Mg and e' m^ P' Mi. 


out of and into view amounts to y^ths of the moon's 

In fig. 79, Plate XYI., a side-view of the moon is 
given. It is supposed to be obtained by rotating the 
moon from the position PE'P'E of fig. 78_, about its 
axis P P' (E approaching)^ or by the observer tra- 
velHng round P P' until he is in the prolongation of 
E. The figure is self-explanatory : but it is to be 
observed that m M m and m' M' m are arcs of 20 32', 
corresponding to the absolute maximum libratory 
swayings, A Gr' and A' G of fig. 78 : p P^ is an arc of 
13 28', corresponding to the maximum libratory 
swaying in latitude (D p D' of fig. 78) ; and e E e is an 
arc of 15 30', corresponding to the maximum swaying 
in longitude {d d' of fig. 78). 

It must always be remembered, however, that 
although such regions as p E p' (fig. 79, Plate XVI.) 
are brought into view by libration, they are always 
seen very much foreshortened, not as presented in 
fig. 79. In fig. 78 the space between the circles jp ejp' e 
and P E P' E' represents the portion of the lunar disc 
within which these regions are always seen; and it is 
easy to see that since their real area is represented by 
the space between the latter circle and the curves p e, 
e p', p' Q 3 and e' p, we can obtain very little insight 
into the configuration of these portions of tie lunar 
surface. A more important efi'ect of the libration is 
to be recognized in the changed aspect under which 
parts within the disc at mean libration are seen at the 
times of maximum libration. Thus the region which 

198 THE moon's changes 

at mean libration occupies tlie portion VE' e p (fig. 
78) of the disc^ is by maximum libration carried to 
the position p e e' tt, with its apparent area three 
times as great, owing to the reduction of foreshorten- 
ing. Thus it can be studied much more favourably. 
Similarly of the four other quadrants. But for this 
circumstance very little value could be attached to 
the portions of many maps representing regions near 
the edge of the disc at the time of mean libration. 

If it is remembered that the time of mean libration 
is also the time when the libratory range is greatest 
for it is only when the mean centre crosses that it 
sways along the arcs AG-' or A'G (fig. 73, Plate XY., 
and fig. 78, Plate XVI.) we see that at such times we 
have the best means for studying the general efiects 
of libration. The last occasion of the kind occurred 
in October, 1871, and at any time within three or 
four months on either side of that epoch libratory 
effects could be studied under favourable conditions. 
The next occasion of the kind will occur in October, 
1874, when the perigee and rising node will be nearly 
in conjunction at the time when the moon is passing 
either.* From what has been already stated it will be 

* The moon will be in perigee on October 25th, at about 6 a.m., 
and at her rising node on October 24th, at about 11-^- p.m., or about 
6|- hours earlier. Thus, at about 3 a.m., on October 2oth, she will 
be very nearly at her mean libration, as she will be " MP' at 7.21 
A.M. of the same day. A very favourable opportunity of studying her 
" mean " aspect will be afforded during the early morning hours of 
October 24th. At 4 h. 36 m. a.m. she enters the penumbra of the 


seen that the moon will be again near her mean 
libration in October 1877, October 1880, and so on, 
for many successive three-yearly intervals. 

This chapter would be incomplete without some 
reference to what has been called the physical libra- 
tion of the moon. 

We have assumed throughout the preceding pages 
that the moon rotates with perfect uniformity on 
her axis, while revolving around the earth. This, 
however, is not strictly the case. In the first place it 
is manifest that since the moon^s mean sidereal revo- 
lution is undergoing at present a process of diminution 
(see pp. 87 90), owing to what is termed her secular 
acceleration, her rotation must either undergo a cor- 
responding acceleration, or she would in the course 
of time so turn round with respect to the earth that 
the regions now unseen would be revealed to terrestrial 
observers. She would, in fact, thus have turned round 
by the time when, owing to her acceleration, she had 
gained half a revolution. It has been shown, however, 
by Laplace, that the attractions to which she is 
subject suffice to prevent such a change, and that her 
rate of rotation changes 'pari passu with her rate of 
revolution. It must, therefore, be to this slight 
extent variable. A similar remark applies to all 
secular perturbations affecting the moon's motions. 
So that it is impossible that the further side of the 

earth, but does not reach the true shadow until 5h. 41*5 m. She 
will be totally eclipsed from 7 till 7h. 32 m., but sets at Greenwich 
before the total phase begins. 

200 THE moon's changes 

moon should ever be turned towards the earth unless 
under the action of some extraneous influence^ as the 
shock of a mass comparable with her own. 

But a real libration much more considerable in 
amount and possibly recognisable by observation, must 
affect the moon's rotation. Newton was the first to 
point out, that if the moon was originally in a fluid 
state, the earth's attraction would draw her into the 
form of a spheroid, the longer axis of which, when 
produced, would pass through the earth's centre. 
" Comparing this phenomenon," says Professor Grant, 
'^ with the tidal spheroid occasioned by the action of 
the moon upon the earth, he found that the diameter 
of the lunar spheroid which is directed towards the 
earth, would exceed the diameter at right angles to 
it by 186 feet. He discovered in this elongation of 
the moon the cause why she always turns the same 
side towards the earth, for he remarked that in any 
other position the action of the earth would not 
maintain her in equilibrium, but would constantly 
draw her back, until the elongated axis coincided in 
direction with the line joining the earth and moon. 
Now, in consequence of the inequalities of the moon 
in longitude, the elongated axis would not always be 
directed exactly to the earth. Newton, therefore, 
concluded that a real libration of the moon would 
ensue, in consequence of which the elongated axis 
would oscillate perpetually on each side of its mean 
place." Thus, if we consider figs. 65, 66, and 67, 
Plate XIV., we see that throughout the motion from 


perigee to apogee the longer axis of the moon is so 
placed that its mean extremity M is on the east of the 
line joining the centre of the earth and moon ; and we 
see from figs. 67, 68 j and 65, that it is on the west 
of that line throughout the motion from apogee to 
perigee. Hence, during the moon's motion from 
perigee to apogee, the earth's attraction tends to re- 
store M to its mean position, or to pull M towards E', 
whereas during the motion from perigee to apogee 
the earth's attraction tends to pull M towards E. As 
the rotation carries M continually in the direction 
E E' (though the revolution prevents us from recog- 
nizing the movement), it follows that as the moon 
moves from perigee to apogee her rotation is accele- 
rated, and as she moves from apogee to perigee her 
rotation is retarded. Thus her rotation rate is at a 
maximum when she is in apogee, and at a minimum 
when she is in perigee. Moreover, her rotation rate is 
above the mean value when she is moving from mean 
distance after apogee to mean distance, and below 
its mean rate as she completes her revolution from 
mean distance after perigee to mean distance again. 
Hence it follows that the greatest real displacement 
of M (or of any given point on the moon's equator) 
occurs when the moon is at her mean distance, and is 
towards E' when the moon is passing to her apogee, 
and towards E when she is moving towards her 
perigee. In other words, the apparent maximum 
libration in longitude is always reduced by this real 

202 THE moon's changes 

Lagrange, in dealing with this relation, noticed 
further, what had apparently escaped Newton's at- 
tention, that owing to the moon's rotation on her 
polar axis, her globe must be, to some slight degree, 
compressed in the direction of this axis. " Lagrange," 
says Professor Grant, " found that both effects were of 
the same order, and that the moon would, in reality, 
acquire the form of an ellipsoid, the greatest axis 
being directed towards the earth, and the least 
being perpendicular to the plane of the equator. 
The greatest and the mean axes will both lie in the 
last-mentioned plane." 

Proceeding to consider the effect of the earth's 
attraction upon the rotating moon, Lagrange found 
that the mean rotation would be affected by a series 
of changes corresponding to those affecting the moon's 
mean motion round the earth. In effect, all the per- 
turbations affecting the moon's motion of revolution 
would, as it were, be reflected, or represented in 
miniature, in these variations of her motion of 

While dealing with this matter, Lagrange noted a 
circumstance to which Newton had not referred, 
though, as Professor Grant well remarks, it is a 
natural corollary to Newton's reasoning. He showed 
that it was not necessary to suppose that the motions 
of revolution and rotation were equal in the beginning. 
If the moon's true rotation once took place in a period 
not absolutely coincident with that of her revolution, 
the attraction of the earth would have sufficed to force 


the rotation-period into mean coincidence with the 
period of revelation. The rotation would, in that 
case, however, no longer be strictly uniform, apart 
from the real libration we have hitherto considered. 
The moon would librate on either side of her mean 
position, independently of her variable motion in her 
orbit. This libration would depend, like the other, 
on the circumstance that .the orb of the moon must be 
somewhat elongated in the direction of the line joining 
the centres of the earth and moon. 

Now the form of real libration last mentioned has 
not been observationally recognized ; but the real 
libration, theoretically predicted by Newton, and con- 
firmed by the analytical researches of Lagrange, has 
been detected by observers. I have said, that in this 
libration every feature of the moon's motions is re- 
flected. Now it might seem, at first sight, that this 
libration would be most noticeable as depending on 
the moon's varying motion in a single revolution, since 
she may be so much as 7 45' before or behind her 
mean place. But, as a matter of fact, the extent 
of the real libration depends much more on the 
length of time during which the earth's action is 
exerted, than on the actual displacement of the 
moon's longer axis from its mean position. Accord- 
ingly, the lunar irregularity called the annual equation, 
although (as we have seen at page 90) it only 
affects the earth's place by a small amount at the 
maximum, yet, as its period is a long one, enables 
the earth to affect the mean rotation rate more effec- 

204 THE moon's changes 

tually than do any of the other hmar perturbations. 
" Bouvard and Nicollet undertook/' says Professor 
Grrant, " a series of careful observations of the moon's 
librations in longitude, at the Royal Observatory 
of Paris. The Connaissance des Temps for 1822 con- 
tains a beautiful paper by Nicollet, in which he sub- 
mitted these observations, amounting in number to 
174, to a searching discussion. The only sensible 
inequality was that corresponding to the annual 
equation. It appeared by observation to have a 
maximum value equal to 4' 45'^ The results at which 
he arrives relative to the ratios of the axes do not 
accord with the generally admitted opinion respecting 
the primitive condition of the moon. He found, in 
fact, that the difference between the least and greatest 
axes was greater than it would be on the supposition 
that the moon was originally a fluid mass." It would, 
however, be rash to base any opinion respecting the 
latter hypothesis upon observations so very delicate 
in their nature. 

It is important to notice that the ellipsoidal form of 
the moon is not only demonstrated by the existence 
of a recognizable real libration, but also by the con- 
tinuance of that singular relation between the position 
of the moon's equator and orbit referred to at p. 174, 
and illustrated in figs. 61 64, Plate XIV. It is 
manifest that since the position of the plane of the 
orbit is continually shifting, this plane would depart 
from coincidence with the plane of the moon's equator, 
unless some extraneous force acted to preserve the 


coincidence. If the moon were a perfect sphere, the 
earth would have no grasp upon her, so to speak, 
whereby to maintain the observed relation between 
the equator plane and the orbit plane. But Lagrange 
has shown that the action of the earth on an ellipsoi- 
dal moon would constantly maintain the coincidence. 
As the coincidence is maintained, we must conclude 
that the moon is necessarily an ellipsoid, and not a 

However, it need hardly be said that no instru- 
mental means at present in our possession could show 
the ellipticity of the lunar disc. Assuming Newton^s 
estimate to be correct, and that the longest axis, 
directed (in its mean position) exactly towards the 
earth, is 186 feet longer than the mean axis E E' of 
the figures in Plate XIY. ; and adopting Lagrange^s 
estimate of the polar compression (as one-fourth of 
the extension of the longest axis), we have the polar 
axis 46 1 feet shorter than the mean axis. Since the 
moon's mean diameter is 2159*6 miles and 46^ feet 
is less than one-113th part of a mile, it follows that 
P P' is less than E E' by less than one-244,000th part 
of either diameter, a quantity altogether inappreciable, 
even independently of the fact that the least of the 
lunar mountains is many times higher than the calcu- 
lated difference between P P' and E E'. 

The ellipsoidal figure of the moon remains none the 
less, however, a demonstrated fact. 




Although the study of the moon^s surface can scarcely 
be said to have been fairly commenced before the in- 
vention of the telescope, yet in very early ages men 
began to form opinions respecting the moon based on 
the appearances presented by her disc. Doubtless the 
ancient Chaldasan, Chinese, Indian, Egyptian, and 
Persian astronomers theorized about the moon's 
physical constitution; but of their views no record 
has reached us. We know only that they studied the 
moon's movements so carefully as to recognize the 
principal features of her orbital motion, but what ideas 
they formed as to the condition of her surface we do 
not know. 

The earliest recorded opinion as to the moon's con- 
dition is the theory of Thales (e.g. 640), that a portion 
of the moon's lustre is inherent. He recognized the 
faint light from the illuminated part of the moon's 
globe at the time of new moon, or rather at the time 
before and after new moon, when the illuminated 
portion forms a narrow crescent; and it was also 
known to him that the moon does not disappear 


wholly when totally eclipsed. He therefore inferred 
that she shines in part by native light. It is some- 
what singular that he did not perceive the remarkable 
contrast which exists between the two kinds of light 
which he regarded as belonging to the moon. The 
deep ruddy colour of the totally eclipsed moon differs 
so completely from the ashy pale light of " the old moon 
in the new moon^s arms,^' that one can hardly under- 
stand how both could be referred to one and the same 
cause. Nevertheless, there have not been wanting 
those who_, in comparatively recent times, have main- 
tained a similar theory. 

Anaxagoras (b.c. 500) was the next of the ancient 
philosophers who theorized respecting the moon. We 
learn from Diogenes Laertius that Anaxagoras regarded 
the moon as an inhabited world, and taught that the 
varieties of tint perceived on her surface are due to 
mountains and valleys. He held and was ridiculed 
for holding the opinion that the moon may be as 
large as the Peloponnesus. 

Some of the Pythagorean philosophers, on the con- 
trary, taught that the moon is a body altogether unlike 
the earth. They regarded her as a smooth, crystalline 
body, having the power of reflecting light like a mirror ; 
and they supposed the spots upon her disc to be the 
reflection of the oceans and continents of our earth. 
But others believed the moon to be an inhabited 
world like the earth, and since daylight on the moon 
continues for about fifteen terrestrial days, they con- 
cluded somewhat boldly that the creatures inhabiting 


the moon must be fifteen times as large as corre- 
sponding terrestrial beings. Heraclitus supposed tlie 
moon to be of tlie same nature as the sun, but darker, 
because involved in the denser part of the earth- 
surrounding ether. Origenes, also maintaining the 
moon to be a self-luminous body, considered her 
surface to be uneven, and regarded the dark spots 
as the shadows of the regions lying higher. 

Passing over many less distinguished names, we 
come to Aristotle, who adopted the theory that the 
light and dark regions in the moon are the reflected 
images of the continents and oceans of our own earth. 
It is worthy of notice that the maintenance of this 
opinion indicates either complete ignorance or a very 
remarkable forgetfulness respecting the laws of re- 
flexion on the one hand, and those relative motions 
of the moon and earth on the other hand respecting 
which even the Ptolemaists held accurate ideas. 
Whether the earth is fixed or in motion, whether 
she rotates or the heavens rotate around her, it is 
certain that her continents and seas are presented in 
a continually varying manner towards the moon. It 
is obvious, then, that if the moon were a mirror 
reflecting the features of the earth, the moon's aspect 
must necessarily change from hour to hour, and from 
day to day. Yet nothing is more certain, even to 
those who only study the moon with the unaided eye, 
that her aspect, so far as the spots on her disc are 
concerned, remains very nearly constant. Her phases 
cause a greater or less portion of her spotted disc to 


be visible to the observer on earthy but the part which 
is seen belongs always to one and the same face. 

The Stoics maintained for the most part that the 
moon is a mixture of fire^ earth, and air, but spherical, 
like the earth and sun. 

Lastly, for it would be idle to devote any con- 
siderable portion of our space to the vague fancies 
which the ancients formed respecting the moon, we 
find that Plutarch strenuously supported the views 
which Anaxagoras had maintained six hundred years 
earlier. He even recognized the indications of moun- 
tains in the moon, in the irregularities of the lunar 
terminator, noting that the lunar mountains would 
necessarily throw vast shadows, precisely as Mount 
Athos, at the time of the summer solstice, cast a 
shadow towards evening which reached across the 
Thracian sea as far as the market-place of Myrina, 
in Lemnos, a distance of eighty-seven miles. 

But it was only after the invention of the telescope 
that just ideas began to be formed as to the condition 
of the moon's surface. 

In May, 1609, Galileo directed towards the moOn 
the first telescope of his own construction. His first 
observations showed him that the moon's surface is 
covered with irregularities; but it was not until he 
applied his largest telescope magnifying only thirty 
times that he recognized the true conformation of 
the lunar surface. He found that the lunar mountains 
are for the most part circular in shape, forming rings 
around depressed regions, and in some respects re- 



sembling tlie mountain-chains which surround Bohe- 
mia. He could perceive bright points of light separated 
by dark spaces from the terminator of the crescent or 
gibbous moon, and he recognized the fact that these 
points are the tops of mountains, illuminated by 
sunlight, while the surrounding valleys are in dark- 
ness. He traced at once the analogy between this 
circumstance and terrestrial phenomena. Those who 
have watched the rising of the sun from the summit 
of a lofty mountain know that when the summit of 
the mountain is in the full glory of sunlight, the sides 
of the mountain are still in shadow, and that the 
neighbouring valleys are plunged in a yet deeper 
gloom. Corresponding appearances are seen when 
the sun is setting. Long before the mountain-tops 
are darkened the level country around is shadowed 
over, and the obscurity of night has already settled 
over ravines and passes. The only difference which 
Galileo perceived in the phenomena of sunrise and 
sunset on the lunar mountains and what is observed 
on our earth, was that no half-lights could be seen, 
nothing but the full blaze of sunlight on the mountain- 
tops and intense blackness in the valleys. Here was 
the first indication of a circumstance on which I shall 
presently have to descant at greater length, the 
absence of any lunar atmosphere, or at least the 
extreme tenuity of whatever air there may be on the 
moon. For it is readily seen that the faint light which 
illuminates the valleys of a mountain- region while as 
yet only the mountain- tops are in sunlight, comes 


from the sky, and the h'ght of the sky is due to the 
existence of an atmosphere. 

The reader will find illustrations of the illumination 
of lunar mountain-tops in the accompanying photo- 
graphs of the moon near her first and third quarter. 

Galileo perceived that in the phenomenon here 
described he possessed the means of measuring the 
altitude of the lunar mountains. Without entering 
into details, it may be remarked that in the case 
of a mountain standing alone on a wide plain, the 
distance of the peak, when just touched by the light, 
from the boundary of light and darkness on the plain, 
depends obviously on the height of the mountain. For, 
in fact, if a person is on the summit of the mountain 
at the moment, he will see the sun on the horizon, and 
the point on his horizon where he sees the sun is in 
reality a point on the plain where also the sun is 
rising at the moment. Now the distance of this 
point, or of the observer's horizon, depends on the 
height of the mountain, as is shown in all our text- 
books of astronomy. Hence, if this distance is 
known, the height of the mountain can be deter- 
mined, and what is true of a mountain on our earth 
is true with certain changes as to details for a mountain 
on the moon. Now it was in Galileo's power to esti- 
mate the apparent distance of a lunar mountain-peak 
in sunlight from the neighbouring terminator, and 
to determine thence the real distance in miles. This 
done, he could estimate the height of the mountain, 
always supposing that the mountain was isolated and 
p 2 


the surrounding region fairly level. Proceeding on 
this assumption, Galileo was led to the conclusion that 
several of the lunar mountains are nearly five miles in 

It will be obvious, however, from a study of the 
moon at her quarters, that this method cannot be 
depended upon alone to give trustworthy results ; 
and this will be yet more manifest to any who will 
examine the moon, when not full, with a telescope of 
even moderate power. It is seen that as a rule not 
only are the lunar mountains not isolated, but the 
surrounding regions are so uneven as to be thrown 
into light or shadow, confusedly intermixed, when the 
sun is low down, that is, when they lie near the ter- 
minator. There is no means of judging exactly where 
the mean terminator lies,< that is, where the boundary 
between light and darkness would lie if the moon 
were a smooth orb. Accordingly very little reliance 
could be placed in the measurements of Galileo, or 
in any estimate of the height of a lunar mountain 
not based on a long and careful study of the region 
surrounding the mountain. 

It is worthy of notice, in passings that the recognition 
of lunar mountains by Galileo was regarded by some 
of his contemporaries as not his least offence against 
the Aristotelian philosophy. Even those who admitted 
that his telescope showed objects which appeared like 
mountains, maintained that in reality the surface of 
the moon is smooth. Over the irregularities perceived 
by Galileo, they argued, there exists a transparent or 


crystalline shell, filling up the cavities and having an 
outer surface perfectly smooth, as Aristotle taught. 
To this argument Galileo gave an answer precisely 
suited to the value of the objection. *' Let them be 
careful," he replied ; " for if they provoke me too far, 
I will erect on their crystalline shell invisible crystal- 
line mountains, ten times as high as any I have yet 

Galileo was the first to recognize the great number 
of craters which exist on certain parts of the moon's 
surface. He compared the craters in the south-western 
quadrant of the moon (see the accompanying lunar 
chart) to the "eyes" in a peacock's tail. 

Galileo's chart of the moon, though creditable to 
him considering his imperfect telescopic means, has 
very little value except as a curiosity. A similar 
remark applies to the researches of Scheiner, Schir- 
laus, and others. 

At this early stage of lunar research the darker 
portions of the moon's surface were considered to be 
seas, the brighter parts being looked upon as land 
regions. Thus we find Kepler saying : " Do maculas 
esse maria, do lucidas esse terras." Galileo himself 
seems to have been better satisfied with his recognition 
of mountains and valleys on the moon than with the 
supposed distinction between land and sea regions. 
It is worthy of notice that in Milton's brief references 
to the Florentine astronomer, based undoubtedly on 
the poet's recollection of his interviews with Galileo 
(see the " Areopagitica"), there is no mention of seas. 


Thus in Book I. of "Paradise Lost^' Milton compares 
the shield of Satan to 

" the moon, whose orb 
Through optic glass the Tuscan artist views 
At evening, from the top of Fesole, 
Or in Val d'Amo, to descry new lands, 
Eivers, or mountains, on her spotty globe." 

Again, in the fifth book, Raphael sees the earth 

" as when by night the glass 
Of Galileo, less assured, observes 
Imagined lands and regions in the moon." 

It is difficult to suppose that Milton would not have 
said " oceans " instead of " regions '^ if Galileo had 
entertained the opinion that the dark lunar regions 
are seas. 

Hevelius, who next made any considerable advance 
in the study of the moon^s surface, adopted Kepler^s 
opinion as to the distinction between the dark and 
bright regions of the moon. He constructed a chart 
which contained more detail and was more correct 
than Galileo's, and adopted a system of nomenclature 
indicating his belief in the existence of analogies 
between lunar and terrestrial regions. Thus we find 
in his list of names, mountains, deserts, marshes, 
seas, lakes, isla,nds, bays, promontories, and straits. 
In some cases he named these from their imagined 
resemblance to terrestrial regions ; in others, he indi- 
cated their appearance as seen in his telescope. Thus, 
the great crater now called Copernicus, was by Hevelius 
called Mount Etna: while the dark enclosed surface 







called Plato was named by Hevelius '' the greater 
black lake/^ The chart by Hevelius was necessarily 
imperfect compared with those now in existence. 
The telescopic power he employed was very little 
greater than that used by Galileo; and he had to 
trust, like Galileo,, to mere estimation of the proportions 
of the different lanar regions, not possessing even the 
roughest appliances for micrometrical measurement. 

Hevelius, following Galileo's method of determining 
the height of the lunar mountains, deduced three miles 
as the maximum height. 

We owe to Hevelius the recognition of the most 
important of the lunar librations. Galileo had detected 
the libration in latitude, and had shown that there 
must also be a small diurnal libration (see last chapter) . 
Hevelius perceived that spots near the eastern aaad 
western edge of the lunar disc were sometimes farther 
on the disc than at others. He not only showed that 
this is due to the libration in longitude, but was able 
to prove that this libration depends on the varying 
motion of the moon and her (appreciably) uniform 

Hevelius's ^^ Selenographia,'' which contains his 
chart (engraved on metal by himself), appeared in 
1647. At this time Peyresl and Gassendi were en- 
gaged in the construction of a lunar chart ; but when 
they heard that Hevelius had completed such a chart 
they ceased from their labour, having drawn only one 
sheet of their chart. 

Father Riccioli, of Bologna, published in 1651 a 


much less valuable chart than that of Hevelius. He 
adopted a new system of nomenclature, replacing the 
terrestrial names of Hevelius by the names of astrono- 
mers and philosophers. Madler says, indeed, that 
Eiccioli's work would have been forgotten had he not 
been led by vanity to find a place for his own name 
on the moon, an arrangement only to be achieved by 
displacing all the names used by Hevelius, at the risk 
of causing perplexity and confusion to later astronomers. 
The charge is rather a serious one. 

Riccioli^s estimates of the altitude of the lunar 
mountains were altogether unsatisfactory. 

Dominic Cassini constructed a chart of the moon 
12 Paris feet in diameter, but not showing many 
details. So far as the method of construction was 
concerned, this map should have been an important 
improvement on its predecessors. The places of the 
chief lunar spots were determined by measurement, 
the other spots were placed by eye-estimates corrected 
for the effects of libration. In 1680 Cassini constructed 
a chart 20 Paris inches in diameter, respecting which 
Madler remarks that it surpassed HeveFs in fulness 
of detail but not in correctness. All the copies of this 
chart were soon sold, and Madler considers it likely 
that the chart was unknown in Germany until a new 
edition was published by Lalande in 1787. 

The first really reliable chart of the moon was con- 
structed by Tobias Mayer. During a lunar eclipse in 
the year 1748, Mayer wished to note the passage of 
the earth^s shadow over the principal lunar features. 


and he recognized the want of an exact chart of the 
moon. It would appear from Lichtenberg^s account, 
that Mayer proposed to himself the construction of a 
chart on a large scale, showing the places of the chief 
lunar spots determined micrometrically. This plan 
he was prevented from carrying out by a pressure of 
other engagements. A small chart, however (7 J Paris 
inches in diameter), was found among his papers, and 
was published at Gottingen in 1775, thirteen years 
after his death, among his ^' Opera inedita,^' and 
remained until 1824 the only trustworthy map of our 

Schroter of Lilienthal studied the moon with great 
care and patience, using first a 7-feet reflector, then 
one of 18 feet, and lastly one of 27 feet in focal length. 
The labours of Schroter as a selenographist were not 
altogether successful, because of his want of skill in 
delineating what he saw. Beer and Miidler consider 
that the accuracy of Schroter's work was further 
affected by his desire to recognize signs of change in 
the moon. But Webb, than whom no better authority 
exists on the subject, says, respecting Schroter's " Se- 
lenographische Fragmente,'^ '^ I have never closed the 
simple and candid record of his most zealous labours 
with any feeling approaching to contempt," and he 
adds that possibly Beer and Miidler were not them- 
selves free from a prepossession opposite to that 
which they condemned in Schroter. 

The work of Lohrmann must be regarded as the 
first really scientific attempt to delineate the moon's 


surface in detail. Lohrmann was a land surveyor 
of Dresden. He planned the construction of a lunar 
chart on a large scale in 25 sections, and in 1824 the 
first four sections were published. But failing* sight 
compelled him to desist from his arduous attempt. 
In 1838 he published an excellent general chart of the 
moon, 15^ inches in diameter.* 

MM. Beer and Miidler began their selenographic 
work in 1830, and their 3-feet chart, together with 
their fine work on the moon,t appeared in 1837. The 
telescope employed by them was only four inches in 
diameter, and the chart does not show every feature 
which can be recognized with a telescope of that 
aperture. Yet the amount of detail is remarkable, 
and the labour actually bestowed upon the work will 
appear incredible to those who are unfamiliar with 
the telescopic aspect of the moon. In " Der Mond,^' 
Beer and Miidler give their measurements of the 
positions of no less than 919 lunar spots, and 1,095 
determinations of the height of lunar mountains.^ 
The map which accompanies the present work was 
reduced by Mr. Webb from the large chart of 
MM. Beer and Madler, and owes no small part 

* Lohrmann died in 1840. 

t "Der Mond, nach seinen kosmischen und individuellen 

X The heights are given in toises, a toise being about 6*3946 
English feet. The highest mountain of all is very appropriately- 
named Newton, and according to the measures of Beer and Madler, 
its summit is 3,727 toises, or about 23,800 feet above the level of 
the floor of the crater. 


of its value to tlie fact that the reduction has been 
made by one who is himself so skilful a student 
of the moon^s surface. The following is Mr. Webb's 
very modest account of a map which has long been 
recognized as a most important contribution to seleno- 
graphy : " It professes to be merely a guide to such 
of the more interesting features as common telescopes 
will reach. It has been carefully reduced from the 
' Mappa Selenographica ' of Beer and Madler^ omitting 
an immense amount of detail accumulated by their 
diligent perseverance, which would only serve to per- 
plex the learner. Selection was difficult in such a 
crowd. On the whole, it seemed best to include every 
object distinguished by an independent name; many 
of little interest thus creep in, and many sufficiently 
remarkable ones drop out; but the line must have 
been drawn somewhere, and perhaps would have been 
nowhere better chosen for the student. Other spots, 
however, have been admitted, from their conspicuous- 
ness, to which Beer and Madler have given only a 
subordinate name ; minuter details come in, in places, 
for ready identification ; elsewhere larger objects are 
passed by, as less useful for the purpose of the map." 

Two lists of the lunar objects in Mr. Webb's map 
first, in the order of the number ; and secondly, in 
alphabetical order will be found at the end of this 
volume. They are identical with those in Mr. Webb's 
" Celestial Objects," third edition. 

I may add that in the year 1869 I carefully examined 
every object included in Webb's map, with a. telescope 


2 5^ Indies in aperture^ using low powers, and satisfied 
myself that the map fulfils in every respect the object 
aimed at by its designer.* 

* I cannot indicate a more pleasing occupation for the possessor 
of a telescope of that size (or any larger size up to four inches) 
than to go over the moon's disc, examining each object seriatim, 
and carefully comparing what is seen with the account given by 
Mr. "Webb. In particular, it is a most useful and instructive 
exercise to observe the varying appearance of particular objects as 
they come into sunlight, as sunlight grows fuller upon them, and 
afterwards, as sunlight passes away from them, until at length they 
are in darkness. The most convenient objects to select for this 
purpose (though it need hardly be said that the true lunarian 
astronomer will not be content with observing these only) are those 
which lie near the terminator of the moon rather early during her 
first quarter, for these will be again on the terminator rather 
early in the third quarter. Thus they can be observed first in the 
early evening, and then later and later, until, when the terminator 
is just leaving them, they must be observed after midnight, but 
not very late ; whereas those objects which are first reached by the 
advancing terminator during the moon's second quarter are left 
by the receding terminator during the fourth quarter, and to be 
well studied at this time must be observed in the early morning 
hours. Those students of astronomy, however, who are ready to 
observe at any hour of the night from twilight to dawn, can study 
any part of the moon from sunrise to sunset at that part. It will 
be obvious that thoroughly to examine any spot on the moon, it 
must be observed during many lunations. Apart from the circum- 
stance that unfavourable weather breaks the continuity of the 
observations, the interval of many hours necessarily elapsing 
between successive observations suffices to render the study of any 
spot during any single lunation imperfect. This is especially the 
case with objects near the eastern and western limbs, because the 
moon must be nearly new (either before or after conjunction with 
the sun) when sunrise or sunset occurs at such points, and the moon 
can only be observed a short time in the morning when she is 
approaching conjunction, and a short time in the evening soon after 





The stereographic map has been constructed by my- 
self from Mr. Webb^s map (as it originally appeared) ; 
it will be found useful for determining the effects of 
foreshortening near the edge of the moon's disc. 

The labours of Schmidt, of Athens, although not as 
yet fully published, must be regarded as altogether 
the most important contribution yet made to seleno- 
graphy. The observations on which the construction 
of his chart has been based were commenced in 1839, 
and in 1865 Schmidt began to combine these observa- 
tions together into chart-form. He proposed at that 
time to have a chart with a diameter of 6 Paris feet, 
and divided into four quadrants, like Miidler's chart. 
The telescope employed for reviewing the observations 
was the refractor of the Athens Observatory, having 

conjunction. But even for other parts of the moon the difficulty 
exists. An observer may watch the progress of sunrise at any spot 
near the terminator of the half-moon, hour after hour, for several 
hom-s in succession ; but he must be interrupted for a much longer 
period, after the moon has approached the horizon too low for useful 
study, until she is again at a fair elevation. Now in the interval 
say sixteen or seventeen hours sunrise or sunset at the spot 
will have made great progress, notwithstanding the great length of 
the lunar day. For sixteen hours on the moon (about a forty-fourth 
part of the lunar day) correspond to more than half an hour on 
the earth, and we know that in every part of the earth the sun's 
place on the heavens alters considerably in half an hour. In fact, 
in sixteen hours, the sun, as seen from the moon, changes his place 
by about eight degrees, and this must importantly affects the 
position and dimensions of the shadows thrown by any lunar 
heights, especially near the time of sunrise and sunset. It is further 
to be considered that the circumstances under which a lunar spot 
is studied vary markedly during the progress of a lunation. 


an aperture of 6 Paris inches. In April^ 1868, the 
work had progressed so far that Schmidt was able to 
form an opinion as to the probable value of a chart 
completed on the adopted plan. He was dissatisfied 
with the result. The work was not exact enough nor 
sufficiently delicate in drawing for his purposes. He 
determined therefore to begin the charting afresh. 
Retaining the original diameter of 6 Paris feet, he 
divided the chart into 25 sections, adopting Lohr- 
mann^s arrangement. Each section forming a much, 
smaller map than the former quadrants, it was 
possible to adopt a much finer and more exact method 
of drawing. He began this work in April, 1868, and 
it is now ready for publication. I believe there are 
difficulties on the score of expense, but these will 
surely be surmounted. When a man has given the 
labours of a life, or the best part of a life, to a scientific 
work of such great difficulty, and with results so 
valuable, it is not asking too much that means should 
be found for publishing the work in a way securing to 
its author a just reward for his untiring exertions. 

The map of Bullialdus and the surrounding region 
affords an idea of Schmidt^s method of delineation. 
It has been reduced, however, considerably from the 
original. The reader should seek out Bullialdus in 
Mr. Webb's map (it is numbered 213, and is in the 
third quadrant). The comparison of the two maps 
will afford an excellent idea of the scale on which 
Schmidt has carried out his processes of charting. 

It remains to be mentioned that a chart of the moon 


is in course of preparation under the supervision of- 
Mr. Birt, and in accordance with a scheme projected 
by the British Association. 

The appHcation of photography to the moon, closely 
associated with the subject of lunar charts, has next 
to be considered. 

So early as 1840 Arago dwelt on the possibility 
that the moon might be persuaded to take her own 
portrait, speaking of the hope that instead of those 
long and wearisome labours by which men had hitherto 
sought to chart the moon, a few minutes might suffice 
to bring her image on Daguerre's prepared plates. 
However, in the very year when Arago made this 
remark, Dr. Draper, of New York, had succeeded in 
photographing the moon. The following history of 
photographic work on the moon is abridged from a 
chapter on the subject contributed by Mr. Brothers 
to Chambers^ " Handbook of Descriptive Astro- 
nomy " : 

" It appears from a paper by Professor H. Draper, 
of New York, published in April, 1864, that in the 
year 1840 his father. Dr. J. W. Draper, was the first 
who succeeded in photographing the moon. Dr. 
Draper states that at the time named (1840) ^it was 
generally supposed the moon^s light contained no 
actinic rays, and was entirely without effect on the 
sensitive silver compounds used in daguerreotyping.^ 
With a telescope of 5 inches aperture Dr. Draper 
obtained pictures on silver plates, and presented 
them to the Lyceum of Natural History of New 


York. Daguerre is stated to have made an un- 
successful attempt to photograph the moon, but I 
have been unable to ascertain when this experiment 
was made. 

" Bond's photographs of the moon were made in 

1 850. The telescope used by him was the Cambridge 
(U.S.) refractor of 15 inches aperture, which gave an 
image of the moon at the focus of the object-glass 

2 inches in diameter. Daguerreotypes and pictures 
on glass mounted for the stereoscope were thus ob- 
tained, and some of them were shown at the Great 
Exhibition of 1851, in London. 

"Between the years 1850 and 1857 we find Secchi 
in Rome, and Bertch and Arnauld in France, and 
in England Phillips, Hartnup, Crookes, De La Rue, 
Fry, and Huggins, appearing as astronomical pho- 
tographers. To these may be added the name of 
Dancer, of Manchester, who in February, 1852, made 
some negatives of the moon with a 4^-incli object- 
glass. They were small, but of such excellence that 
they would bear examination under the microscope 
with a 3 -inch objective, and they are believed to be 
the first ever taken in this country. Baxendell and 
Williamson, also of Manchester, were engaged about 
the same time in producing photographs of the moon. 

" The first detailed account of experiments in 
celestial photography which I have met with is by 
Professor Phillips, who read a paper on the subject 
at the meeting of the British Association at Hull in 
1853. In it he says: ' If photography can ever succeed 


in portraying as mucli of tlie moon as the eye can see 
and discriminate^ we shall be able to leave to future 
times monuments by which, the secular changes of the 
moon's physical aspect may be determined. And if 
this be impracticable if the utmost success of the 
photographer should only produce a picture of the 
larger features of the moon, this will be a gift of the 
highest value, since it will be a basis, an accurate and 
. practical foundation of the minuter details, which, with 
such aid, the artist may confidently sketch/ The 
pictures of the moon taken by Professor Phillips were 
made with a 6|-inch refractor, by Cooke, of 11 feet 
focus: this produced a negative of I5 inch diameter 
in 30 seconds. Professor Phillips does not enter very 
minutely into the photographic part of the subject, 
but he gives some very useful details of calculations 
as to what may be expected to be seen in photographs 
taken with such a splendid instrument as that of Lord 
Rosse. It is assumed that an image of the moon may 
be obtained direct of 12 inches diameter, and this, when 
again magnified sufficiently, would show ^ black bands 
12 yards across.' What may be done remains to be 
seen, but up to the present time these anticipations 
have not been realized. 

'' We have next, from the pen of Crookes, a paper 
communicated to the Royal Society of London in 
December, 1856, but which was not read before that 
Society until February in the following year. Mr. 
Crookes appears to have obtained good results as 
early as 1855, and, assisted by a grant from the 



Donation Fund of tlie Eoyal Society^ he was enabled 
to give attention to the subject during the greater 
part of the year following. The details of the pro- 
cess employed are given with much minuteness. The 
telescope used was the equatorial refractor at the 
Liverpool Observatory, of 8 inches aperture and 
12^ feet focal length, which produced an image of 
the moon 1*35 inch diameter. The hody of a small 
camera was fixed in the place of the eye-piece, so that 
the image of the moon was received in the usual way 
on the ground glass. The chemical focus of the 
object-glass was found to be y%ths of an inch beyond 
the optical focus, being over- corrected for the actinic 
rays. Although a good clock movement, driven by 
water-power, was applied to the telescope, it was 
found necessary to follow the moon^s motion by 
means of the slow-motion handles attached to the 
right ascension and declination circles, and this was 
effected by using an eye-piece, with a power of 200 
on the finder, keeping the cross-wires steadily on one 
spot. With this instrument Hartnup had taken a 
large number of negatives, but owing to the long 
exposure required he was not successful; but with 
more suitable collodion and chemical Solutions, and 
although the temperature of the Observatory was 
below the freezing-point, Mr. Crookes obtained dense 
negatives in about 4 seconds. Crookes afterwards 
enlarged his negatives 20 diameters, and he expresses 
his opinion that the magnifying should be conducted 
simultaneously with the photography by having a 


proper arrangement of lenses^ so as to throw an en- 
larged image of the moon at once on the collodion 
plate ; and he states that the want of light could be 
no objection J as an exposure of from 2 to 10 minutes 
would not be ' too severe a tax upon a steady and 
skilful hand and eye/ 

" In an appendix to his paper Mr. Crookes gives 
some particulars as to the time required to obtain 
-negatives of the moon with different telescopes, from 
which it appears that the time varied from 6 minutes 
to 6 seconds. The different results named must, I 
conclude, have been caused not so much by the differ- 
ences in the instruments as in the various processes 
employed, and in the manipulation. I must observe, 
also, that it is not stated whether all the experiments 
were tried upon the full moon a point materially 
affecting the time. 

^^ In 1858 De La Rue read an important paper 
before the Royal Astronomical Society, from which it 
appears that the light of the moon is from 2 to 3 times 
brighter than that of Jupiter,* while its actinic power 
is only as 6 to 5, or 6 to 4. On Dec. 7, 1857, Jupiter 
was photographed in 5 seconds and Saturn in 1 minute, 
and on another occasion the moon and Saturn were 
photographed in 15 seconds just after an occultation of 
the planet. 

"The report of the Council of the Royal Astrono- 

* Theoretically the light of the moon should be nearly 27 times 
as bright as Jupiter's, since Jupiter is 5^ times farther from the 


mical Society for 1858 contains the following re- 
marks : ^ A very curious result^ since to some extent 
confirmed by Professor SeccM, has been pointed out 
by De La Rue^ namely, that those portions of the 
moon's surface which are illumined by a very oblique 
ray from the sun possess so little photogenic power 
that, although to the eye they appear as bright as 
other portions of the moon illumined by a more direct 
ray, the latter will produce the effect, called by photo- 
graphers solarization, before the former (the obliquely- 
illumined portions) can produce the faintest image/ 
And the report also suggests that the moon may 
have a comparatively dense atmosphere, and that 
there may be vegetation on those parts called 

'' At the meeting of the British Association at 
Aberdeen, in 1859, De La Rue read a very valuable 
paper on Celestial Photography. An abstract of it was 
published at the time in the British Journal of Photo- 
graphy, and in August and September of the following 
year further details of this gentleman's method of 
working were given in the same journal. The pro- 
cesses and machinery employed are so minutely de- 
scribed that it is unnecessary here to say more than 
that he commenced his experiments about the end of 
1 852, and that he used a reflecting telescope* of his 

* " The advantage of the reflecting over the refracting telescope 
is very great, owing to the coincidence of the visual and actinic 
foci ; but it will presently appear that the refractor can be made 
to equal, if not excel, the work of the reflector." 


own manufacture of 13 inches aperture and 10 feet 
focal lengthy which gives a negative of the moon 
averaging about yV*^ ^^ ^^ inch in diameter. The 
photographs were at first taken at the side of the tube 
after the image had been twice reflected. This was 
afterwards altered so as to allow the image to pass 
direct to the collodion plate, but the advantage gained 
by this method was not so satisfactory as was expected. 
In taking pictures at the side of the tube, a small 
camera box was fixed in the place of the eye-piece, and 
at the back a small compound microscope was at- 
tached, so that the edge of a broad wire was always 
kept in contact with one of the craters on the moon^s 
surface, the image being seen through the collodion 
film at the same time with the wire in the focus of 
the microscope. This ingenious contrivance, in the 
absence of a driving-clock, was found to be very 
effectual . and some very sharp and beautiful negatives 
were thus obtained. De La Rue afterwards applied 
a clockwork motion to the telescope, and his negatives 
taken with the same instrument are as yet the best 
ever obtained in this country. 

" Nearly a quarter of a century has elapsed since 
the moon was first photographed in America, and a 
good deal has been done since on that side of the 
Atlantic. To an American we are indebted for the 
best pictures of our satellite yet produced, and it is 
difficult to conceive that anything superior can ever 
be obtained ; and yet with the fact before us that De 
La Rue's are better than any others, taken in this 


country^ so it may prove that even the marvellous 
pictures of Mr, Rutherfurd may be surpassed. 

" Mr. Rutherfurd appears, from a paper in Silli- 
man's American Journal of Science for May, 1865, tO' 
have begun his work in lunar photography in 1858 
with an equatorial of 11^ inches aperture and 14 feet 
focal length, and corrected in the usual way for the 
visual focus only. The actinic focus was found to be 
y^oths of an inch longer than the visual. The instru- 
ment gave pictures of the moon, and of the stars down 
to the fifth magnitude, satisfactory when compared 
with what had previously been done, but not suffi- 
ciently so to satisfy Mr. Rutherfurd, who, after trying 
to correct for the photographic ray by working with 
combinations of lenses inserted in the tube between 
the object-glass and sensitive plate, commenced some 
experiments in 1861 with a silvered mirror of 13 inches 
diameter, which was mounted in a frame and strapped 
to the tube of the refractor. Mr. Rutherfurd enume- 
rates several objections to the reflector for this kind 
of work, but admits the advantage of the coincidence 
of foci. The reflector was abandoned for a refractor 
specially constructed, of the same size as the first one, 
and nearly of the same focal length, but corrected 
only for the chemical rays. This glass was completed 
in December, 1864, but it was not until March 6th of 
the following year that a sufficiently clear atmosphere 
occurred, and on that night the negative was taken 
from which the prints were made.^' 

Mr. Brothers has himself taken many photographs 



of the moon witli great success,, though using a tele- 
scope (refracting) only live inches in aperture. The 
convenience and simplicity of the arrangements he 
employed will be recognized when it is mentioned 
that on the evening of the partial eclipse of the moon, 
Oct. 4, 1865, he succeeded, " with the help of two as- 
sistants, in taking no less than 20 negatives, though 
the telescope was several times disturbed to oblige 
friends who desired to see the progress of the eclipse 
through the instrument.''^ 

Before passing to the description of the general 
results which have followed from the telescopic obser- 
vation of the moon, as well as from processes of 
charting and photographing, it will be well to discuss 
the observations which have been made on the moon^s 
light viz., first on the total quantity of light which 
she reflects, when fall, towards the earth ; secondly, 
on the varying proportion of light so reflected when 
she is at her phases; and thirdly, on the different 
light-reflecting qualities of different portions of her 

The consideration of the total quantity of light 
reflected by the moon implies" in reality the question 
what degree of whiteness she possesses. For a per- 
fectly white object* would reflect all the light it 
received, but a coloured object reflects only a portion, 

* There is no such thing as perfect whiteness in nature (refer- 
ring to opaque objects). Even new-fallen snow does not reflect so 
much as four-fifths of the incident light. The following table 
(resulting from the observations of Zollner) is useful for purposes 



while a perfectly black object would reflect none. 
An object of many colours and the moon un- 
questionably is such an object may be said to tend 
(as a whole) towards blackness or whiteness, accord- 
ing as it reflects less or more of the light which shines 
upon it. 

Let us first consider the comparison between the 
moon's light and the sun's, according to the best 
observations hitherto made : 

The observations of Bouguer assigned to the moon 
a total brightness equal to one 300,000th part of the 
sun's. The method he employed was the direct com- 
parison between sunlight and candlehght, and between 
moonlight and candlelight. Wollaston also took 
candlelight as the means of comparison, but deter- 
mined the relative brightness of the sources of light 

of comparison, the total light incident on a surface being represented 

by unity : 

Snow just fallen reflects 


. 0783 

White paper 


. 0-700 

White sandstone 


. 0-237 

Clay marl 


. 0-156 

Quartz porphyry 

. 0-108 

Moist soil 

. 0-079 

Dark grey syenite 

. 0-078 

These objects shine by diffused r< 

jflected light. 

For light regu 

larly reflected the following table is 

useful : 

Mercury reflects 


. 0-648 

Speculum metal 


. 0-535 



. 0-040 



.. 0-032 



. 0-021 


by the method of equalizing the shadows. He obtained 
the result that the moon's light is but one 801,070th 
part of the sun's. 

We owe, however, to ZoUner the most satisfactory 
determination of the moon's total brightness. He 
employed two distinct methods. In one he determined 
the illumination by comparing surface-brightness ; in 
the other he obtained point-like imp^ges of the sun 
and moon for comparison with corresponding images 
of candle-flames. The results obtained by these two 
methods were in close agreement, according to one, 
the light of the full moon is one 618,000th part of the 
sun's light, while, according to the other, the propor- 
tion is as 1 to 619,000. 

It would be easy to determine from this result the 
exact proportion of the incident light which the 
moon's surface, regarded as a whole, is capable of re- 
flecting, if the moon were a smooth but unpolished 
sphere : for we know exactly what proportion of the 
sun's light the moon intercepts, and it is also known 
that a smooth half-sphere seen under full illumination, 
reflects two-thirds of the light which a flat round disc 
of the same diameter would reflect. But the problem 
is complicated in the present instance by the uneven- 
ness of the moon's surface, which causes the light to 
fall upon various parts of the lunar surface at angles 
very diff'erent from those in the case of a smooth 
sphere. In fact, it is perfectly manifest from the 
aspect of the full moon, that we have to deal with a 
case very different from that of a smooth, or, as it is 
called, mat surface. For such a surface, seen as a 


disc under full solar illumination^ would be brightest 
at the centre, and shade off gradually to the edge ; 
whereas it is patent to observation that the disc of the 
full moon is as bright near the edge as near the 
centre.* Before we can undertake the inquiry, there- 
fore, into the moon's average brightness, we must 
endeavour to ascertain what effect should be ascribed 
to the inequalities upon her surface. 

This has been accomplished by Zollner in a suffi- 
ciently satisfactory manner, by comparing the total 
quantity of the moon's light at her various phases, 
with what would be obtained if the moon were a smooth 
sphere. It is obvious that as the different parts of the 
moon's disc, when she is full, do not shine with the 
brightness due to a smooth surface, we might expect 
to find her total brightness at any given phase mar- 
kedly different from the value estimated for the case 
of a smooth sphere. This Zollner found to be the 
case. The ' full ' moon is far brighter by comparison 
with the gibbous moon (especially when little more 
than half full), than would happen if she were smooth. 
Now the considerations on which Zollner based his 
interpretation of this peculiarity are not suited to 

* It is necessary to exercise some caution, however, in adopting 
a result of this kind, since the eye is very readily deceived. We 
see the full moon on a dark background, and this certainly tends 
to add to the apparent brightness of the edge of the disc. As a 
case illustrating this effect of contrast, it may be mentioned that 
Jupiter appears to the eye to be brighter near the edge than near 
the middle of the disc, and yet when his disc is examined with a 
graduated darkening glass, it is found to be brighter near the middle 
than near the edge. 



these pages, involving analytical considerations of 
some complexity. The result, therefore, is all that 
need here be stated. Zollner^s conclusion is, that the 
average slope of the lunar inequalities amounts to 
about 52 degrees.* Be it noticed, that this result is 

" The following table will show how closely the results obtained 
by Zollner agreed with the empirical formula which he deduced 
from his estimate of the mean slope of the lunar irregularities. 
The first column gives the distance of the moon from full, the 
distance being regarded as positive when the observation was made 
after the time of full moon, and as negative when the observation 
preceded full moon : 

Theoretical Brightness. 

Arc from Moon's 

Full Moon'8 as 100. 


place to poiat 



tbe Sua. 

Moon regarded 

By Zollner' s 


as smooth. 


+ 1 














" 84-04 






















+28 ' 
















. +40 




















+ 58 













in no degree affected by observations of the apparent 
slope of lunar mountains and craters, because irregu- 
larities much smaller than any which the telescope can 
detect, would suffice to explain the observed variations 
of brilliancy. If the whole surface of the moon were 
covered with conical hills only a foot, or even only an 
inch, in height, the same general result would be 
produced as though there were mountains of the same 
form a mile, or several miles, in height. 

It appears from this result, that the brightness of 
the full moon is considerably greater than it would 
be if the moon were a smooth sphere ; and, in fact, 
Zollner would seem to regard the brightness of the 
full moon as very nearly equal to that of a flat disc of 
equal diameter. I do not enter here into a calculation 
of the quantity of light which such a disc would reflect ; 
but the following result may be accepted as sufficiently 
near the truth. A perfectly white disc of the same 
diameter as the moon^s, and under direct solar illu- 
mination, would have a total brightness equal to about 
one 92,600th part of the sun^s. Now we have seen 
that the actual quantity received from the moon is 
about one 618,000th part of the sun^s light; and 
taking into account the smaller mean disc of the 
moon, as compared with the sun, we find that the 
moon's light is rather more than one-sixth part of 
that of a disc of perfect whiteness, under direct 
solar illumination, and looking as large as the 
moon's disc. Zollner deduces from his estimate 
of the mean irregularity of the moon's surface. 


a result so near to this as to imply what I have 
just stated, viz._, that he regards the brightness of 
the full moon as not much less than that of a flat disc 
of equal size, and having a surface of the same average 
reflective power. For he sets the light of the full 
moon as rather less than a sixth part of that which 
it would have if the moon were made of a perfectly 
white substance. The exact proportion assigned by 
him is that of 1,736 to 10,000. This is what, following 
Lambert, he calls the albedo, or whiteness of the 
moon, and he justly remarks that, considering her 
whole brightness, she must be regarded as more nearly 
black than white. Nevertheless, he adds that from 
his estimates of the moon's brighter parts he is satis- 
fied that their whiteness can be compared with that 
of the whitest of terrestrial substances.* 

It is worthy of notice that Sir John Herschel had 
already in a far simpler way deduced a result closely 
agreeing with ZoUner's. It will be seen from the 
table in the note at p. 232 that white sandstone reflects 
about 0*237 of the incident light; and it may be 
inferred from other values in that table that weathered 
sandstone rock would have an albedo of about 0*150. 
Now Herschel remarks that the actual illumination 
of the lunar surface is not much superior to that of 
weathered sandstone rock in full sunshine. ^' I have 
frequently ,'' he proceeds, "compared the moon 

* His words are : " Dass der Mond an seinen helleren und hell- 
8ten Stellen aus einem Stoffe besteht, der, auf die Erde gebracht 
zu dem weisesten der uns bekannten Korper gezahlt werden wiird," 


setting behind tlie grey perpendicular fa9ade of the 
Table Mauntain, illuminated by the sun just risen in 
the opposite quarter of the horizon, when it has been 
scarcely distinguishable in brightness from the rock 
in contact with it. The sun and moon being nearly at 
equal altitudes, and the atmosphere perfectly free 
from cloud or vapour, its effect is alike on both 

A difficulty will present itself to most readers on a 
first view of Zollner's result. The full moon, taken 
as a whole, appears white when high above the 
horizbn on a dark clear night ; and it appears quite 
impossible to regard her as more nearly black than 
white. Again, as another form of the same difficulty, 
it appears obvious to any one who regards ordinary 
sandstone or any substance of like reflective power 
in full daylight, that the brightness of the substance 
is markedly inferior to that of the full moon at mid- 
night. Herein is illustrated one of those effects of 
contrast which are so deceptive in all questions of 
relative brightness. We see the full moon in a dark 

* It is to be noted, however, that the illumination of the sand- 
stone would be reduced by atmospheric absorption, which would 
not happen, of course, with the moon. The effect of atmospheric 
absorption in reducing the apparent brightness of the moon thus 
fully illuminated, and of the sandstone thus not quite fully illu- 
minated, would not be equal, because the sandstone was seen 
through only a portion of the atmospheric strata interposed 
between the eye and the moon. Hence would result a near 
approach to equalization so far as atmospheric eflfects are con- 


background, and with no other object comparable to 
her in brightness,, and the eye accordingly overesti- 
mates her light, a comparison is made between her 
real but not obvious partial blackness, and the very 
obvious and much greater blackness of the sky, and 
thus the idea of whiteness is suggested. On the con- 
trary, when we look at stone or rock illuminated by 
full sunlight, objects as brightly, or even more brightly, 
illuminated are all around, and the eye accordingly 
estimates fairly, or perhaps even underestimates, the 
whiteness of the illuminated rock.* 

* Amongst hundreds of illustrations of the effect of contrast in 
deceiving the eye in such cases (a subject of the utmost importance 
in astronomical observations) may be mentioned our estimate of 
the brightness of the old moon in the new moon's arms. Nothing 
can be more certain than that in reality the light of the old moon 
in this case is due to illumination by the earth, and at a moderate 
computation this illumination exceeds full moonlight twelve times. 
(The only doubtful point is the average light-reflecting quality of 
the earth's surface, which I am here assuming to be rather less 
than that of the moon's surface.) Now we know how bright a 
landscape appears when bathed in full moonlight, and we can 
infer that under twelve times that amount of light the brightness 
would be very considerable. Assuredly an object as large in 
appearance as the moon would under such light appear very con- 
spicuous, and white. Yet the old moon in the new moon's arms, 
though illuminated to this degree, can scarcely be perceived at 
all until twilight has made some progress. The light of the 
early evening sky is quite sufficient to render the considerable light 
of the old moon quite imperceptible. To this may be added the 
fact that the disc of the moon during total eclipses, although it 
appears so dark to the eye, is nevertheless illuminated by nearly 
full earth-light, and certainly with ten times the lustre of a 
terrestrial landscape under full moonlight. 


Eeturning to Zollner^s remark tliat the brightest 
parts of the moon are comparable in whiteness with 
the whitest terrestrial substances, it follows obviously 
that the darker portions of the moon are very much 
less bright than the average. Thus the bright 
summit of Aristarchus, whose reflective power is so 
great that, as seen on the dark part of the moon 
(when therefore only illuminated by earth-light), it has 
been mistaken for a volcano in eruption, has probably 
a reflective power equal to that of new-fallen snow, or 
0*783, which exceeds the average whiteness of the 
moon about 4| times. And we may assume that the 
dark floor of Plato and the yet darker Grimaldi are as 
far below the average of brightness. But even dark- 
grey syenite, the lowest in reflective power of all the 
substances in ZoUner^s table (see note, p. 235), reflects 
0*078 of the incident light, which indicates a white- 
ness nearly half the average whiteness of the moon's 
surface. We may safely assume that the darkest 
parts of the moon are blacker than this. 

A question of much greater diSiculty is suggested 
by observations which appear to indicate changes in 
the brightness of certain lunar regions. Some obser- 
vations of this kind are referred to in a subsequent 
chapter. At present I shall merely remark that such 
observations do not appear to have been hitherto 
made in such a way as to afibrd convincing evidence 
that change takes place with the progress of the lunar 
day. In particular, it seems to me that the readiness 
with which the eye may be deceived by the effect of 


contrast has not been duly taken into account. There- 
fore, while recognizing, in the observations directed 
to the recognition of tint-changes or colour- changes, 
a possible means of advancing in a very marked 
manner our knowledge of the moon's condition, I find 
myself at present unable to regard as demonstrated 
any of the phenomena which are described by those 
who have made researches in this department of 

In considering the general results of the telescopic 
scrutiny of the moon, it is well to remember the cir- 
cumstances under which such scrutiny has been 

The highest power yet applied to the moon (a 
power of about six thousand) brings her, so to speak, 
to a distance of forty miles, a distance far too great 
for objects of moderate size to become visible. Many 
of my readers have probably seen Mont Blanc from the 

* I have been supplied by Mr. Nelson, F.R.A.S., with a series of 
very interesting observations made by him, which tend to show that 
the Floor of Plato darkens with the advance of the lunar day and 
grows lighter as the day wanes, confirming observations of a similar 
character made and collected by Mr. Birt. But a careful study 
of these observations, as well as of all the observations of the same 
kind that I have had access to, has not satisfied me that the whole 
series of phenomena may not be subjective merely. No sufficient 
precautions have been hitherto taken to eliminate as far as possible 
the effect of contrast, and till this has been done it would be unsafe 
to adopt any conclusion as demonstrated. Moreover, the assumption 
that the floor is smooth seems to me altogether unsafe. "We cannot 
possibly be certain that it is not covered with irregularities too 
small to be individually discernible. 


neighbourhood ofGreneva, a distance of about forty miles. 
At this distance the proportions of vast snow-covered 
hills and rocks are dwarfed almost to nothingness, ex- 
tensive glaciers are quite imperceptible^ and any attempt 
to recognize the presence of living creatures or of their 
dwellings (with the unaided eye) is utterly useless. 
But even this comparison does not present the full 
extent of the difficulties attending the examination 
of the moon^s surface with our highest powers. The 
circumstances under which such powers are applied 
are such as to render the view much less perfect than 
the mere value of the magnifying power employed 
might seem to imply. We view celestial objects 
through tubes placed at the bottom of a vast aerial 
ocean, never at rest through any portion of its depth ; 
and the atmospheric undulations which even the naked 
eye is able to detect are magnified just in proportion 
to the power employed. These undulations are the 
bane of the telescopist. What could be done with 
telescopes, if it were not for these obstructions to 
perfect vision may be gathered from the results of 
Professor SmytVs observations from the summit of 
Teneriffe. Raised here above the densest and most 
disturbed strata, he found the powers of his telescope 
increased to a marvellous extent. Stars which he had 
looked for in vain with the same instrument in Edin- 
burgh now shone with admirable distinctness and 
brilliancy. Those delicate stipplings of the discs of 
Jupiter and Saturn, which require in England the 
powers of the largest telescopes, were clearly seen in 


the excellent but small telescope lie employed in Ms 
researches. It is probably not too mucb to say that 
even if the Rosse telescope were perfect in defining 
power, which unfortunately is very far indeed from 
being the case, yet on account of atmospheric disturb- 
ance, instead of reducing the moon's distance to 
forty miles, it would in fact not be really effective 
enough to reduce that distance to less than 150 miles. 

Accordingly, though we recognize in the grey 
plains or seas on the moon the appearance of smooth- 
ness, it is very far from being certain that these 
regions may not in reality be covered with irregu- 
larities of very considerable slope. The assumption 
that they resemble old sea-bottoms, or that in smooth- 
ness they are analogous to deserts and prairies on our 
own earth, seems an unsafe one. The uniformity of 
curvature which marks their surfaces ^s a whole does 
indeed afford an argument in favour of their having 
once been in a liquid condition ; but that their solidi- 
fication should have resulted in a smooth surface is far 
from being certain. On the contrary, it seems not 
unlikely that the true surface may be marked with 
corrugations, or crystalline formations, or other uni- 
form unevennesses, if one may so speak. 

It is a noteworthy circumstance that the lunar 
plains do not form portions of the same sphere, 
some lying deeper than others, that is, belonging to 
a sphere of smaller radius. 

Again, it is to be remembered that the mountain- 
chains on the moon are seen under circumstances 
E 2 


which enable us to recognize none but the boldest 
features of these formations. It is as unsafe to 
theorize as to their geological or selenological confor- 
mation, as it would be to speculate on the structure 
of a mountain-range on earth which had only been seen 
from a distance of two or three hundred miles. The 
following description by Mr. Webb must be read with 
this consideration carefully held in remembrance : 
" The mountain-chains/' he remarks,, " are of very 
various kind : some are of vast continuous height 
and extent, some flattened into plateaux intersected 
by ravines, some rough with crowds of hillocks, some 
sharpened into detached and precipitous peaks. The 
common feature of the mountain- chains on the earth 
a greater steepness along one side is very perceptible 
here, as though the strata had been tilted in a similar 
manner. Detached masses and solitary pyramids are 
scattered here and there upon the plains, frequentty 
of a height and abruptness paralleled only in the most 
craggy regions of the earth. Every gradation of cliff and 
ridge and hillock succeeds; among them a large number 
of narrow banks " (that is, of banks which look narrow 
at the enormous distance from which they are seen), 
" of slight elevation but surprising length, extending 
for vast distances through level surfaces : these so 
frequently form lines of communication between more 
important objects, uniting distant craters or moun- 
tains, and crowned at intervals by insulated hills, that 
Schroter formerly, and Beer and Miidler in modern 
times, have ascribed them to the horizontal working 


of an elastic force, which^ when it readied a weaker 
portion of the surface,, issued forth in a vertical up- 
heaval or explosion. The fact of the communication/' 
he justly adds, " is more obvious than the probability 
of the explanation." 

But although, as will be manifest from the photo- 
graphs which illustrate this work, the lunar moun- 
tain-ranges form by no means an unimportant feature 
of the moon's surface, the crateriform mountains 
must be regarded as the more characteristic feature. 
If w^e adopt Mallet\s theory of the formation of surface- 
irregularities on a planet, we must assume that the 
intermediate stage between the formation of great 
elevated and depressed regions corresponding to our 
continents and oceans, and the epoch of volcanic 
activity, lasted but a relatively short time. If the 
crateriform mountains were due to volcanic action, 
then that action must have lasted longer, must have 
been more widespread, and must have been also far 
more intense than on our own earth. The considera- 
tions thus suggested are discussed in a subsequent 
chapter. Here I shall consider only the classification 
of the crateriform mountains. They may be con- 
veniently divided, after Webb, into walled or bul- 
warked plains, ring mountains, craters, and saucer- 
shaped depressions or pits. " The second and third,'' 
he remarks, " differ chiefly in size ; but the first have 
a character of their own, in the perfect resemblance of 
their interiors to the grey plains, as though they had 
been originallv deeper, but filled in subsequently with 


the same material^ many of them, in fact, bearing 
evident marks of having been broken down and over- 
flowed from the outside. Their colour is often sug- 
gestive of some kind of vegetation, though it is 
difficult to remark this with the apparent deficiency of 
air and water. It has been ingeniously suggested 
that a shallow stratum of carbonic acid gas, the 
frequent product of volcanoes and long surviving their 
activity (for instance, among the ancient craters of 
Auvergne, where it exists in great quantity), may in 
such situations support the life of some kind of 
plants ; and the idea deserves to be borne in mind in 
studying the changes of relative brightness in some of 
these spots. The deeper are usually the more concave 
craters; but the bottom is often flat, sometimes con- 
vex, and frequently shows subsequent disturbance in 
ridges, hillocks, minute craters, or more generally, as 
the last eff'ect of eruption, central hills of various 
heights, but seldom attaining that of the wall, or even, 
according to Schmidt, the external level. The ring is 
usually steepest within, as in terrestrial craters, and 
many times built up in vast terraces, frequently lying, 
Schmidt sajs, in pairs divided by narrow ravines. 
Nasmyth refers these not very probably to suc- 
cessively decreasing explosions; in other cases he 
more reasonably ascribes them to the slipping, down 
of materials upheaved too steeply to stand, and 
undermined by lava at their base, leaving visible 
breaches in the wall above. They would be well 
explained on the supposition of fluctuating levels in 


a molten surface. Small transverse ridges occasion- 
ally descend from the ring, cliiejfly on the outside ; 
great peaks often spring up like towers upon the 
wall; gateways at times break through the rampart, 
and in some cases are multiplied till the remaining 
piers of wall resemble the stones of a huge megalithic 

The accompanying picture of CopernicuSj* taken 
by Father Secchi with the fine refractor of the 
Roman Observatory, aptly illustrates the appearance 
of large craters when seen with powerful telescopes. 
I give Mr. Webb's description of the crater in full, as 
showing his method of dealing with lunar details, in 
the admirable work to which I have already invited 
the reader's attention at p. 219. " Copernicus,'^ he 
says, "is one of the grandest craters, 56 m. in 
diameter. It has a central mountain (2,400 feet in 
height, according to Schmidt), two of whose six 
heads are conspicuous; and a noble ring composed 
not only of terraces, but distinct heights separated 
by ravines ; the summit, a narrow ridge, not quite 
insular, rises 11,000 feet above the bottom, the 
height of Etna; after which Hevel named it. Schmidt 
gives it nearly 12,800 feet, with a peak of 13,500 
feet, west ; and an inclination in some places of 60. 
Piazzi Smyth observed remarkable resemblances 

* The cut is one of the large number of engravings illustrating 
Pr. Secchi's book on the sun now in my hands, aud under process 
of translation. It has been kindly lent to me by Messrs. Longmans 
for the illustration of the present work. 



between the interior conchoidal cliffs and those of the 
great crater of Tenerifife. 

A mass of ridges leans 

The Lunar Crater Copernicus (Secchi). 


upon the wall, partly concentric, partly radiating : 
the latter are compared to lava. The whole is beauti- 
fully, though anonymously, figured in Sir J. HerschePs 
' Outlines of Astronomy/ There is also a large 
drawing by Secchi''^ (from which the accompanying 
picture has been reduced) ; ^' but this grand object 
requires, and would well reward, still closer study. 
It comes into sight a day or two after the first 
quarter. Vertical illumination brings out a singular 
cloud of white streaks related to it as a centre. It is 
then very brilliant, and the ring sometimes resembles 
a string of pearls. Beer and Miidler once counted 
more than fifty specks. '' 

Schmidt^s map of BuUialdus and the neighbourhood 
also well illustrates the nature of the lunar crateri- 
form mountains of various dimensions. But yet 
further insight into the characteristics of the more 
disturbed and uneven portions of the moon^s surface 
will be obtained from the study of Plate XX., which 
represents a very rough and volcanic portion of the 
moon^s surface, as modelled from telescopic observa- 
tions by Mr. Nasmyth. The engraving was taken 
from a photograph of the original model furnished to 
Sir J. Herschel by Mr. Nasmyth ; and I am indebted 
to Messrs. Longmans for permission to use this 
admirable engraving in the present work. 

^^A succession of eruptions may be constantly 
traced,^^ Mr, Webb remarks, " in the repeated en- 
croachment of rings on each other, where, as Schmidt 
says^ the ejected materials seem to have been disturbed 


before they had time to harden, and the largest are 
thus pointed out as the oldest craters, and the gradual 
decay of the explosive force, like that of many terres- 
trial volcanoes, becomes unquestionable. The peculiar 
whiteness of the smaller craters may indicate some- 
thing analogous to the difference between the earlier 
and later lavas of the earth, or to the decomposition 
caused, as at Teneriffe, by acid vapours in the grey 
levels. We thus perhaps obtain an indication of the 
superficial character of their colouring.^' 

The lunar valleys include formations as remarkable 
as the long banks described above, viz., the clefts or 
rills, furrows extending with perfect straightness for 
long distances, and changing in direction (if at all) 
suddenly, thereafter continuing their course in a 
straight line. These were first noticed by Schroter, 
and a few were discovered by Gruithuisen and Lohr- 
man ; but Beer and Madler added greatly to the known 
number, which was raised by their labours to 150. 
Schmidt has discovered nearly 300 more. Mr. Webb 
makes the following remarks on the rills: "These 
most singular furrows pass chiefly through levels, in- 
tersect craters (proving a more recent date), reappear 
beyond obstructing mountains, as though carried 
through by a tunnel, and commence and terminate 
with little reference to any conspicuous feature of the 
neighbourhood. The idea of artificial formation is 
negatived by their magnitude (Schmidt gives them 
18 to 92 miles long, \ to 2-^ miles broad) : they have 
been more probably referred to cracks in a shrinking 


LorLdo]i:LorLfirLa_ns frCo. 


surface. The observations of Kunowsky, confirmed 
by Dorpat^ seem in some instances to point 
to a less intelligible origin in rows of minute con- 
tiguous craters ; but a more rigorous scrutiny with the 
highest optical aid is yet required/^ 

A feature which is well seen in the illustrative pho- 
tographs, but best in the view of the full moon, is the 
existence of radiating streaks from certain craters. 
The most remarkable system of rays is that which has 
the great crater Tycho. as its centre. It will be seen 
from the photograph of the full moon that this system 
can be recognized over a very large proportion of the 
visible hemisphere, and doubtless extends on the south 
(that is, the uppermost part of the picture) far upon 
the unseen hemisphere. The same picture well exhibits 
the radiating bright streaks from Copernicus, Kepler, 
and Aristarchus j and three other less striking systems 
can be recognized in this view of the moon. The 
telescope shows others. '' In some cases," Webb re- 
marks, '^the streaks proceed from a circular grey 
border surrounding the crater ; in others they cross 
irregularly at its centre. They pass alike over moun- 
tain and valley, and even through the rings and 
cavities of craters, and seem to defy all scrutiny^' 
(query, interpretation). Nichol makes the following 
suggestive comments on this peculiarity, though in 
quoting his remarks I would not be understood to 
imply assent to them in all respects : "They consist of 
broad brilliant bands (seen in their proper splendour 
only when the moon is full) issuing from all sides of 


the crater, and stretching to various distances from 
their origin, one of them can be traced along a reach 
of 1,700 miles. There are several defining characteris- 
tics of these bands. First, It is only when the moon 
is full that we see them in their entire clearness. They 
may be traced, although very faintly, when the moon 
is not full : their splendour at full moon is very great. 
This cannot wholly be attributed to the effect of direct 
instead of oblique Hght, because at the edges of the 
moon^s apparent disc, on which the solar ray falls 
very obliquely at full moon, their brilliancy is the 
same. No rational explanation whatever has been 
proposed regarding this remarkable peculiarity. 
Secondly, The light thrown towards us by the rays 
from Tycho is of the same kind as that reflected from 
the edge and centre of the crater itself; so that the 
matter of which they are composed had probably the 
same origin as those other portions of Tycho. Thirdly, 
These rays pass onward in thorough disregard of the 
general contour of the moon's surface ; nowhere being 
turned from their predetermined course by valley, 
crater, or mountain-ridge. Now, this critical fact 
quite discredits the hypothesis that they are akin to 
lava, or that they are merely superficial. A stream 
of lava spreads out on meeting a valley or low-land, 
and forms a lake ; nor can it ever overpass a mountain 
barrier. The question remains then, are these rays 
composed of matter that has been shot up from the 
interior of the moon? It may seem incredible that 
we can solve this problem by virtually digging pits of 


vast depth down through those singular bands, and 
thus ascertaining practically that the matter composing 
them certainly descends towards the interior of our 
satellite, and that in all probability it has been forced 
up from that interior. The telescope, which in this 
instance is our labourer, has discovered numerous 
small craters of varying depth in the midst of many 
of the rays, and it reveals the fact, that these small 
craters, however deep, do not penetrate through the 
matter we are examining, inasmuch as there comes 
from their bases always the same kind of light that 
characterizes the ray. There is one remarkable case 
in point. A large crater named Saussure, and not far 
from Tycho, lies directly in the line of a ray, and of 
course appears to interrupt it ; but at the bottom of 
Saussure, notwithstanding the great depth of that 
crater, the ray from Tycho may be traced. Nay, there 
is reason to believe that in favourable circumstances 
the same ray might be seen rising up the sides of 
Saussure, just as a vein of trap or of volcanic rock 
pierces the sedimentary strata upon earth. What, 
then, can we make of such phenomena ? Are not our 
terrestrial trap dykes or veins their fitting similitudes ? 
Piercing the other rocks, as if shot up from below, 
these singular veins pass onward across valley and 
over mountain ; their direction their own independent 
for the most parfc of the rocks they have cut ; they 
appear, too, in systems, some limited in magnitude, 
and evidently radiating from a known source ; others 
of vast extent, and usually considered parallel, but 


probably owing their apparent parallelism to the fact 
that we trace them only through a brief portion of 
their course. Accept this analogy^ and none other 
appears within reach^- and the rays or bright lines of 
the moon assume an import quite unexpected, they 
become indices to those successive dislocations that con- 
stitute epochs in the progress of our satellite." Elsewhere 
Nichol indicates in what sense he uses these words : 
where any system of radiations is intersected by 
another, it is manifest that the later formation will 
alone have its radiations unbroken at the places of 
intersection. Then Nichol assigns as the result of 
the telescopic scrutiny of the radiations from Coper- 
nicus, Aristarchus, and Kepler, that the three systems 
were formed in the order in which they are here named. 
He also assigns to the radiations from Tycho (mani- 
festly with good reason) a great antiquity. " Another 
indication,^^ he proceeds, " furnished by the rays 
demands notice. Reflect on the course, as to con- 
tinuous visibility, of any stream of lava or any trap 
dyke upon the surface of the earth. No lava current 
from Etna could be traced to any great distance by 
a spectator in the moon, however powerful his tele- 
scope ; and it would be the same with regard to those 
lines or dykes of trap, even supposing them endowed 
with an excessive, power to reflect light. The reason 
is that they soon enter forest regions, and are con- 
cealed there, or become overspread by grass or other 
vegetable carpetings. But not even a lichen stains 
the brightness of the bands issuing from Tycho ; they 


preserve, not their visibility merely, but one invariable 
brightness through their entire courses. The in- 
ference is but too clear ; and we are glad to find a 
refuge from it, in the certainty that arrangements 
must be different on the other face of our satellite. 
The existence of a rocky desert, devoid of life or living 
thing, of the extent of even one lunar hemisphere, is 
startling enough. '' 

Nasmyth is of opinion that the radiations ^' are 
cracks divergent from a central region of explosion, 
and filled up with molten matter from beneath.-" But 
Webb objects that this theory is irreconcilable with the 
fact that the radiations seldom, if ever, cause any 
deviation in the superficial level. '' Trap dykes on the 
earth are indeed apt to assume the form of the surface, 
but the chances against so general and exact a resto- 
ration of level all along such multiplied and most 
irregular lines of explosion, would be incalculable; 
many of the rays are also far too long and broad for 
this supposition, or for that of Beer and Madler, that 
they may be stains arising from highly heated subter- 
ranean vapour on its way to the point of its escape." 
It appears to me impossible to refer these phenomena 
to any general cause but the reaction of the moon^s 
interior overcoming the tension of the crust; and to 
this degree Nasmyth^s theory seems correct ; but it 
appears manifest also, that the crust cannot have been 
fractured in the ordinary sense of the word. Since, 
however, it results from Mallet^s investigations that 
the tension of the crust is called into play in the earlier 


stages of contraction, and its power to resist pressure 
in the later stages, in other words, since the crust 
at first contracts faster than the nucleus, and afterwards 
not so fast as the nucleus, we may assume that the 
radiating systems were formed in so early an era that 
the crust was plastic. And it seems reasonable to 
conclude that the outflowing matter would retain its 
liquid condition long enough (the crust itself being 
intensely hot) to spread widely, a circumstance which 
would account at once for the breadth of many of the 
rays, and for the restoration of level to such a degree 
that no shadows are thrown. It appears probable 
also, that not only (which is manifest) were the craters 
formed later which are seen around and upon the radia- 
tions, but that the central crater itself acquired its 
actual form long after the epoch when the rays were 
formed. In the chapter on the moon^s physical con- 
dition, considerations will be dealt with which bear 
upon this point. At the moment I need only remind 
the reader that the processes of cooling must have 
proceeded much more rapidly in the moon^s case than 
in the earth's, and that this circumstance serves to 
account for phenomena indicativeof a widely extended 
bursting of the lunar crust. I am disposed to believe, 
moreover, that although the radiating systems are 
manifestly not contemporaneous, they were all formed 
during a period of no great duration possibly, in- 
deed, not lasting for more than a few years, if so 

The following peculiarities of arrangement noted by 


Mr. Webb should be carefully noted in connection 
with the considerations dealt with in Chapter VI. 
" The remarkable tendency to circular forms, even 
where explosive action seems not to have been con- 
cerned, as in the bays of the so-called seas, is very 
obvious; and so are the horizontal lines of communi- 
cation. The gigantic craters, or walled plains, often 
affect a meridional arrangement : three huge rows of 
this kind are very conspicuous near the centre, and 
the east and west limbs. A tendency to parallel 
direction has often a curious influence on the position 
of smaller objects : in many regions these chiefly point 
to the same quarter, usually north and south, or north- 
east and south-west ; thus in one vicinity (between G, 
L, and M, in the map of the moon). Beer and Miidler 
speak of thirty objects following a parallel arrange- 
ment for one turned any other way ; even small craters 
entangled in such general pressures (as round L) have 
been squeezed into an oval form, and the effect is like 
that of an oblique strain upon the pattern of a loosely- 
woven fabric: an instance (near 27, 28 on the map) 
of double parallelism, like that of a net, is mentioned, 
with crossing lines from south-south-west and south- 
east. Local repetitions frequently occur ; one region 
(between 290 and 292) is characterized by exaggerated 
central hills of craters ; another (A) is without them ; 
in another (185) the walls themselves fail. Incomplete 
rings are much more common towards the north than 
the south pole ; the defect is usually in the north, 
seldom in the . west part of the circle ; sometimes a 


cluster of craters are all breached on the same side 
(near 23, 32). Two similar craters often lie north and 
south of each other, and near them is frequently 
a corresponding duplicate. Two large craters occa- 
sionally lie north and south, of greatly resembling 
character the southern usually three-fourths of the 
northern in size from 18 to 36 miles apart, and con- 
nected by ridges pointing in a south-west direction 
(20, 19; 78, 77; 83, 84; 102, 103; 208, 207, 204; 
239, 242 ; 261, 260 ; 260, 263 ; 340, 345). Several of 
these arrangements are the more remarkable, as we 
know of nothing similar on the earth.''^ 

But, interesting as these observations may be, it 
has not been for such discoveries as these that astro- 
nomers have examined the lunar surface. The exami- 
nation of mere peculiarities of physical condition is, 
after all, but barren labour, if it lead to no discovery of 
physical condition. The principal charm of astronomy, 
as indeed of all observational science, lies in the study 
of change, of progress, development, and decay, and 
specially of systematic variations taking place in re- 
gularly-recurring cycles. The rings of Saturn, for 
instance, have been regarded with a new interest, 
since the younger Struve first started the theory of 
their gradual change of figure. The " snowy poles of 
moonless Mars,^' in like manner, have been examined 
with much more attention and interest by modern 
astronomers than they were by Cassini or Maraldi, 
precisely because they are now recognized as snow- 
covered regions, increasing in the Martial winter and 


diminishing in the Martial summer. In this relation the 
moon has been a most disappointing object of astro- 
nomical observation. For two centuries and a half, her 
face has been scanned with the closest possible scru- 
tiny; her features have been portrayed in elaborate 
maps ; many an astronomer has given a large portion 
of his life to the work of examining craters, plains, 
mountains, and valleys for the signs of change ; but 
hitherto no certain evidence or rather no evidence 
save of the most doubtful character has been afforded 
that the moon is other than " a dead and useless waste 
of extinct volcanoes. ^^ 

Early telescopic observations of the moon were con- 
ducted with the confident expectation that the moon 
would be found to be an inhabited world, and that 
much would soon be learned of the appearance and 
manners of the Lunarians. With each increase of 
telescopic power a new examination was conducted, 
and it was only when the elder HerschePs great re- 
flector had been applied in vain to the search, that 
men began to look on the examination as nearly hope- 
less. Herschel himself, who was too well acquainted, 
however, with the real difficulties of the question to 
share the hopes of the inexperienced, was strongly of 
opinion that the moon is inhabited. After describing 
the relations, physical and seasonal, prevailing on the 
lunar surface, he adds, '''there only seems wanting, 
in order to complete the analogy, that it should be 
inhabited hke the earth.^' 

When Sir John Herschel conveyed a powerful re- 
s 2 


fleeter to Cape Town, tlie hope was renewed tliat 
something might yet be learned of the hinar in- 
habitantSj through observations conducted in the 
pure skies of the southern hemisphere. So con- 
fidently was this hope entertained and expressed, 
that the opportunity seemed a good one to some 
American wits to play off a hoax on those who were 
anxiously awaiting the result of Sir John^s observations. 
Accordingly an elaborate account was prepared of a 
series of discoveries respecting the appearance and 
behaviour of certain strange and not very well-con- 
ducted creatures inhabiting the moon. The readiness 
with which the story was believed in many quarters 
was a sufficient indication of the prevalence of the 
opinion that the moon is inhabited. 

Lord Rosse^s giant reflector has been applied, as 
we have said, to the examination of the moon^s sur- 
face, without any results differing in character from 
those already obtained. 

The considerations discussed at p. 242 are sufficient 
to show that it is not only hopeless to look on the 
moon^s surface for the presence of living creatures, 
but even to look for constructions erected by such 
supposed inhabitants of the moon, unless these works 
were far greater than the largest yet constructed by 
man. Large cities, indeed, might be visible, but not 
separate edifices; nor could variations in the dimensions 
of cities be easily detected. It has been argued, 
indeed, that since gravitation, which gives weight to 
living creatures as well as to the objects around them. 


is SO much less at tlie moon's surface than at the earth's, 
lunar inhabitants might, without being cumbrous or 
unwieldy, be very much larger than the races sub- 
sisting on our earth; they might also easily erect 
buildings far exceeding in magnitude the noblest works 
of man. Nor is the argument wholly fanciful. A man 
of average strength and agility placed on the lunar 
surface (and supposed to preserve his usual powers 
under the somewhat inconvenient circumstances in 
which he would there find himself) could easily spring 
four or five times his own height, and could lift with 
ease a mass which, on the earth, would weigh half a 
ton. Thus it would not only be possible for a race 
of lunarians, equal in strength to terrestrial races, to 
erect buildings much larger than those erected by 
man, but it would be necessary to the stability of lunar 
dwellings that they should be built on a massive and 
stupendous scale. Further, it would be convenient 
that the lunarians, by increased dimensions and more 
solid proportions, should lose a portion of the super- 
abundant agility above indicated. Thus we have at 
once the necessity and the power for the erection of 
edifices far exceeding those erected by man. 

But having thus shown that lunar structures might 
very possibly be of such vast dimensions as to become 
visible in our largest telescopes, it remains only to 
add, that no object that could, with the slightest 
appearance of probability, be ascribed to the labours 
of intelHgent creatures, has ever been detected on the 
moon's surface. 


Failing the discovery of living creatures^ or of their 
works, it was hoped that at least the telescope might 
reveal the progress of natural processes taking place 
on a sufficiently important scale. There can hardly 
be a doubt that our earth_, examined from the moon^s 
distance, would exhibit (in telescopes of considerable 
power) a variety of interesting changes. It would be 
easy to trace the slow alternate increase and diminution 
of the polar snow- caps. The vast llanos, savannahs, 
and prairies would exhibit with the changing seasons 
very easily distinguishable changes of colour; the 
occasional covering of large districts by heavy snow- 
falls would also be a readily recognizable phenomenon. 

Now the moon^s surface exhibits distinctly-marked 
varieties of colour. We see regions of the purest 
white regions which one would be apt to speak of 
as snow-covered, if one could conceive the possibility 
that snow should have fallen where (now, at least) 
there is neither air nor water. Then there are the so- 
called seas, grey or neutral-tinted regions, differing 
from the former not merely in colour and in tone, but 
in the photographic quality of the light they reflect 
towards the earth. Some of the seas exhibit a greenish 
tint, as the Sea of Serenity and the Sea of Humours. 
Where there is a central mountain, within a circular 
depression, the surrounding plain is generally of a 
bluish steel-grey colour. The region called the Marsh 
of Sleep exhibits a pale red tint, a colour seen also 
near the Hercynian mountains, within a circumvalla- 
tion called Lichtenburg. 


But although there are varieties^ there has never 
yet been detected any variation of colour. Nothing 
has been seen which could be ascribed,, with any 
appearance of probability^ to the effects of seasonal 

Failing evidence of the existence of living creatures^ 
or of processes of vegetation^ there only remains one 
form of variation to be looked for : I refer to changes 
such as^ on our own earth, are produced by volcanic 
eruptions or by earthquakes. 

It is evident, in the first place, that the inquiry 
must be one of extreme delicacy. Indeed, if the 
volcanic changes taking place on the moon were no 
greater than those observed on our own earth, it 
would be almost hopeless to seek for traces of their 
existence. The light proceeding from a burning 
mountain could never be detected at the moon^s 
distance. It would also be extremely difiicult to 
detect such small new craters as have been formed 
on our earth. It is the overspreading of extensive 
tracts with the materials ejected from volcanoes that 
would form the most readily detected feature of change. 
There have been instances in which, for many miles 
around a volcano, the country has been covered with 
ashes, and there can be little doubt that the change 
of appearance thus produced might be detected even 
at the moon^s distance. There have also, been cases 
in which, during an earthquake, the features of an 
extensive region have been entirely altered. Instances 
such as these, however, are so few and far between^ 


that if we supposed the moon^s surface correspond- 
ingly altered, the chances would be great against the 
detection of such change. 

Assuming the probable, or, at least, the possible, 
existence of active volcanoes upon the moon, it remains 
to be seen how the operation of such volcanoes is to 
be detected from our earth. The colours seen in dif- 
ferent parts of the moon's surface are little marked, 
and grey or neutral-tinted regions are so prevalent 
that it would be very difficult to note the change of 
colour produced by the downfall, over large tracts, of 
matter ejected from erupting volcanoes. Differences 
of elevation produced by such downfalls afford a much 
more favourable object of examination. 

One of the earliest to record the supposed occur- 
rence of volcanic action upon the moon was the elder 
Herschel. He observed luminous appearances, which 
he attributed to the presence of active volcanoes on 
the dark part of the moon's disc. The cause of these 
(which had been noticed also, but less satisfactorily, 
by Bianchini and Short) has now been shown to be 
the greater brilliancy of the light reflected under par- 
ticular circumstances from our own earth upon the 

* The following statements by Prof. Shaler, of the Harvard 
University, afford interesting evidence respecting the degree of 
illumination of the " old moon in the new moon's arms " ; 
" With the 15-inch Merz of the Observatory of this university it 
is possible, imder favourable conditions, to see all the principal 
features of the topography on the dark region illuminated only by 
this earth-shine. In the course of some years of study upon the 


Schroter, who devoted a large part of his time to 
the study of the moon, imagined that he had detected 
signs of change, which, singularly enough, he seems 

geology, if we may so call it, of the moon, I have had several 
opportunities of seeing under these conditions all the great features 
of the dark surfice shine out with amazing distinctness. The curious 
point, however, is that the eye is not enabled to recognize the craters 
by light and shade, for the light is too feeble for that, besides 
being too vertical for such a result ; but the relief is solely due to 
the difference in the light-reflecting power of the various features 
of the topography. Whatever becomes very brilliant under the 
vertical illumination of the full moon (the edges and floors of many 
craters, certain isolated hills, and the radiating bands of light) 
shines out with a singular distinctness when lit by our earth's 
light. This is important, inasmuch as it shows pretty conclusively 
that the difference in the brightness of various parts of the surface 
of the moon is not due to the effects of the heating of the surface 
during the long lunar day, but is dependent upon difference in the 
light-reflecting power. There are several degrees of brightness ob- 
servable in the different objects which shine out by the earth-light. 
In this climate there are not over three or four nights in the year 
when the moon can be caught in favourable conditions for this 
observation. The moon should not be over twenty-four hours old 
(the newer the better), and the region near the horizon should be 
reasonably clear. Under these conditions I have twice been able 
to recognize nearly all the craters on the dark part, over 15 miles 
in diameter, and probably one-half the bands, which show with a 
power of 100 when the moon is full. That this partial illumination 
of the dark part of the moon is in no way connected with the 
'action of an atmosphere, is clearly shown by the fact that the light 
is evenly distributed over the whole surface, and does not diminish 
as we go away from the part which is lit by direct sunlight, as it 
should do if an atmosphere were in question. It will be noticed 
that this fact probably explains the greater- part of the perplexing 
statements concerning the illumination of certain craters before the 
terminator came to them. It certainly accounts for the volcanic 


to have been disposed to attribute rather to changes 
in a lunar atmosphere of small extent than to volcanic 
action. He was not able to assert positively^ however, 
that appreciable changes had taken place. One ob- 
servation of his, however, deserves special notice, as 
will presently appear. In November, 1 788, he noted 
that the place of the crater Linnaeus, in the Sea of 
Serenity, was occupied by a dark spot, instead of 
appearing, as usual, somewhat brighter than the 
neighbouring regions. Assuming that he made no 
mistake, we have evidence here of activity in this 
particular crater. 

Since the time of Schroter, other observers have 
been led to suspect the occurrence of change. Mr. 
Webb pointed out in 1865 eight noteworthy instances. 
Several of these seem easily explained by the well- 
known effects of difference in telescopic powers, ob- 
servational skill, keenness of vision, and the like ; but 
there are one or two which seem to deserve a closer 
scrutiny : 

On February 8, 1862^ the south-south-west slope of 
Copernicus was seen to be studded with a number of 
minute craters not seen in Beer and Madler's map. 
These seemed to form a continuation of a region 
crowded with craters between Copernicus and Era- 

actinty which has so often been supposed to be manifested by 
Aristarchus. Under the illumination of the earth-light this is by 
far the brightest object on the dark part of the moon's face, and is 
visible much longer and with poorer glasses than any other object 


tosthenes. And it is singular tliat this last-named 
region exhibits a honeycombed appearance^ which 
appears not to have existed in Schroter's time, since 
it is not recorded in his maps, and could hardly have 
escaped his persevering scrutiny. 

Another instance of supposed change is well worthy 
of attention, as showing the diflBculty of the whole 
subject. There is a ring-mountain, called Mersenius, 
which has attracted the close attention of lunar ob- 
servers, in consequence of its convex interior a very 
uncommon feature. This bubble-like convexity is 
represented by Schroter, and also by Beer and 
Madler, as perfectly smooth. Not only is this the 
case, but we have distinct evidence that Beer and 
Madler paid particular attention to this spot. Now, 
in 1836, only a year or two after the publication of 
Beer and Madler^s map, Mr. Webb detected a minute 
crater on the summit of the convexity within Mer- 
senius; he also saw several delicate markings, re- 
sembling long irregular ravines, " formed by the 
dropping-in of part of an inflated and hollow crust.^^ 
Here one would feel satisfied that a change had taken 
place, were it not that in Lohrman's map a minute 
crater had already been inserted on the convexity in 
question, while from the dates (1822 and 1836) 
between which Lohrman constructed his map, the 
probability is that the crater had been observed by 
him at or before the time when Beer and Madler failed 
to detect its existence. 

I have already referred to Schroter's observations 


of the crater Linne on the Sea of Serenit3\ Whether 
Schroter had been deceived or not, when he asserted 
that a dark spot hid the place of this crater in 1788, 
it is certain that during the last half-century the crater 
had been distinctly visible. When the sun is high 
upon Linne, it appears as a small bright spot, but 
when the spot is near the terminator, the crater has 
been recognizable through the appearance of a shadow 
within and without its wall. Now, in October, 1866, 
Schmidt observed that the crater Linne had disap- 
peared. When the spot was close to the terminator 
no shadow could be seen, as usual, either within or 
without the crater. In November he again looked in 
vain for Linne. 

It is to be noted that the crater is no insignificant 
formation, but fully five and a half miles wide, and 
very deep. It is, in fact, one of the largest craters 
within the Sea of Serenity; (H in WebVs map, 
where Linne is numbered 74) . 

The crater is represented in Riccioli^s map (pub- 
lished in 1653). We have seen, also, that in 1788 
Schroter recorded the appearance of a dark spot, 
instead of a crater, in Linnets place. Lohrman, in 
1823, observed Linne to be one of the brightest 
spots on the whole surface of the moon. His map 
shows Linne as a distinct crater, and he describes 
it as more than one (German) mile in diameter, 
very deep, and visible in every illumination. In 
Beer and Madler's map the crater is also distinctly 
noted ; they measured its position no less than 


seven times; and they describe it as very deep and 
very bright. In photographs by De La Rue and 
Rufcherfurd, Linne appears as a very bright spot ; 
but singularly enough^ we have also photographs 
taken during the month in which Schmidt looked in 
vain for the crater^ and in these photographs (taken 
by Mr. Buckingham _, of Walworth) Linn4 though dis- 
cernible as a light spot, has scarcely one-third of the 
brilliancy observed in De La Rue^s and Rutherfurd^s 
photographs, taken between the years 1858 and 1865. 

Mr. Webb, one of our most careful observers, 
examined the Mare Serenitatis on December 13, 1866, 
for confirmation or disproof of Schmidt's views. The 
following is extracted from his notes of this observa- 
tion : '' About one-third of the way from a marked 
high mountain on the northern shore of the Sea of 
Serenity, is a minute darkish-looking crater. This I 
presume to be Linne, as I can trace no crater anywhere 
ehe. At some little distance south-east, there is an ill- 
defined ivhitishness on the floor of the sea." When 
Mr. Webb tested the results of his observations by 
means of a lunar map, he found that the first-named 
crater was not Linne, and that the '^' ill-defined 
whitishness '' occupied the exact spot on which Linne 
is depicted. Subsequent observations fully confirmed 
the existence of this spot, which, singularly enough, 
is found, on careful measurement, to be twice as large 
as the crater whose place it conceals. 

Many other observers who carefully examined Linne 
agreed in confirming the results of Schmidt's obser- 


vation. One of the most satisfactory observations of 
Linne was effected by Father Secchi at Eome. On 
the evening of February 10, 1867_, he watched Linne 
as it entered into the sun^s hght, and on the 11th he 
renewed his observations. In place of the large crater 
figured in lunar maps, he could just detect with the 
powerful instrumental means at his command a very 
small crater, smaller even than those craters which 
have received no names. ^' There is no doubt/' he 
said, " that a change has occurred.'^ Schmidt, it may 
be mentioned, independently detected the small crater 
described by Secchi. 

The evidence thus far was as follows : Where 
there used to be a ring-mountain surrounding a deep 
crater so that, under suitable illuminations, the 
shadow of the mountain could be seen within and 
without the crater no shadow could be traced; a 
space, considerably larger than that originally sur- 
rounded by the ring-mountain, appeared somewhat 
brighter than the neighbouring parts of the Sea of 
Serenity ; in very powerful telescopes a minute black 
spot could be seen in place of the original wide and 
deep crater. It seemed clear, then, first, that there 
had not been a mere eruption of ashes filling up the 
crater, because then we should still see the shadow of 
the ring-mountain. Nor could the whole region have 
sunk, because then a large shadow would appear when 
the spot was near the terminator. The ring-mountain 
had not been destroyed, because its fragments and 
their shadows would remain visible. The only ex- 


planation available^ therefore, appeared to be this, 
that a mass of matter had been poured into the crater 
from below, and had overflowed the barrier formed by 
the ring-mountain, so as to cover the steep outer sides 
of the ring. Instead, therefore, of an outer declivity 
which could throw a shadow, there appeared to be an 
inclination sloping so gradually that no shadow could 
be detected, the whole surface thus covered with 
erupted matter shining with the same sort of light, so 
that a spot was seen somewhat lighter than the Sea of 
Serenity, and larger than the original crater. 

Not only did the above explanation account for all 
the observed appearances, but it corresponded to 
phenomena of eruption presented on our own earth. 
Mud volcanoes (or salsen), as distinguished from vol- 
canoes proper, present a very close analogy to the 
process of change just described. '' Mud volcanoes,^' 
says Humboldt, ^^ continue in a state of repose for 
centuries. When they burst forth, they are accom- 
panied by earthquakes, subterranean thunder, the 
elevation of a whole district, and (for a short time) by 
the eruption of lofty flames. After the first forcible 
outburst, mud volcanoes present to us the picture of 
an incessant but feeble activity.'^ 

Yet subsequent observations have not confirmed 
the interpretation thus placed on the apparent changes 
in Linne. It has been shown by several observers, 
and notably by Mr. Browning in 1867, that Linne 
changes remarkably in aspect in a very short space of 
time, under changing solar illumination; and the in- 


ference would seem to be_, that the supposed changes 
have been merely optical. Many observers of expe- 
rience still retain the opinion, however, that there has 
been a real change in this region. 

In Chapter VI. reasons will be suggested for behov- 
ing that, owing to the changes of temperature of the 
moon's surface, as the long lunar day and night suc- 
ceed each other, gradual processes of change must 
take place in the surface-contour. 

The history of the inquiries which have been 
made as to the actual heating of the moon's surface 
during the lunar day is full of interest, but in this 
place I must be content with a brief account of the 

There are two ways in which the moon's surface 
sends out heat towards the earth. First, a portion of 
the sun's heat must necessarily be reflected precisely 
as the sun's light is reflected. But the moon's surface 
must also be heated by the sun's rays, and this heat 
is radiated into space. Thus at and near the time of 
full moon, the moon's surface is reflecting sun-heat 
towards us, and it is also giving out the heat which it 
has itself acquired under the sun's rays. Now the 
distinction between these two forms of heat is recog- 
nizable by instrumental means. The reflected heat is 
of the same quality as direct solar heat, and accord- 
ingly passes readily, like sun-heat, through absorbing 
media, such as glass, moist air, and others, which have 
the power of preventing the passage of heat which is 
merely radiated from bodies not so far heated as to 


become highly luminous.* We see this fact illustrated 
in our greenhouses. The sun^s heat passes freely 
through the glass (at least only a small proportion is 
prevented from passing), but the warmed interior of 
the greenhouse does not part thus freely with its heat, 
the glass preventing the heat from passing away. 
Accordingly, when evening comes on, the interior of 
the greenhouse becomes considerably warmer than the 
surrounding air. In like manner, the heat reflected 
by the moon will pass freely through glass, while the 
heat which she radiates will not so pass ; and in this 
circumstance we recognize the means of comparing the 
quantity of heat which the moon reflects and radiates, 
and thus of determining the degree to which the 
moon^s surface is actually heated at any given time. 

The first inquiries made into this subject did not, 
however, deal with relations so delicate as these. 
" Probably/^ says the anonymous writer of a fine essay 
on the subject in Fraser's Magazine for January, 1870, 
" the old observers had exaggerated notions of the 
moon's warmth and thought they could measure it by 
an ordinary thermometer. This was the tool employed 

"" We may state the matter thus : the shorter' heat-waves pass 
through the media in question, the longer heat-waves are absorbed. 
From researches by Dr. Draper, it may be inferred that heat is 
produced, not merely or chiefly by waves from the red end and 
beyond the red end of the spectrum, but by waves from all parts 
of the visible spectrum and from beyond both ends of the spectrum. 
His researches, as also those of Sorby of Sheffield, demonstrate also 
that chemical action is produced by sether-waves of all orders of 


by one Tschirnausen^ who condensed the moon-ligh 
by means of burning glasses, in hope of getting mea- 
surable warmth, somewhere about the year 1699. Of 
course he got nothing. The famous La Hire followed 
suit some half a dozen years after, using a three-foot 
burning mirror and the most delicate thermometer 
thenknown; he, too, could obtain no indication, though 
his mirror condensed the light, and any heat with it, 
some 300 times ; that is to say, the quantity of light 
falling upon the reflector was concentrated upon a spot 
one-three-hundredth of its area. After these failures, 
a century elapsed, and then Howard, and subsequently 
Prevost, attempted to gain direct evidence of lunar 
caloric, but since they had only expansion thermo- 
meters at their command, their results were valueless ; 
for one, from some accidental circumstance, brought 
out a temperature obviously too high, while the other 
found negative heat ! '' 

The much more effective heat-measuring instrument 
called the thermopile, was first brought into action by 
Melloni. Space does not permit me to describe here 
at length the nature of this instrument, for a full de- 
scription of which I would refer the reader to Prof. 
TyndalFs " Heat considered- as a Mode of Motion.-'^ 
Suffice it to say that the heat to be measured is 
suffered. to fall on the place of junction of plates of 
bismuth and antimony, and that the electric current 
thus established is measured by the movement of a 
delicately poised magnetic needle. Melloni " concen- 
trated the lunar rays '^ (says the account from which 


I have already quoted) " by means of a metallic 
mirror, upon the face of his thermopile, in the hope 
of seeing the needle swing in the direction indicating 
heat ; but it turned the opposite way, proving that 
the anterior and exposed surface of the pile was colder 
than its posterior face. Here was an anomaly. Did 
the moon, then, shed cold ? No, the reverse action 
was due to the frigorific effect of a clear sky : the pile 
cooled more rapidly on one side than on the other, and 
a current was generated by this disturbance of the 
thermal equilibrium ; a current, however, of opposite 
character to that which would have been produced if 
the moon had rendered the exposed face of the pile 
warmer than that which was turned away from the 
sky. Melloni^s experiments were made about the 
year 1831. 

" Two or three years after this the late Professor 
Forbes set about some investigations upon the polar- 
ization of heat, which involved the use of a very sen- 
sitive thermopile, and he was tempted to repeat 
Melloni^s moon-test, with the substitution of a lens 
for a mirror as a condenser. The diameter of this 
lens was 30 inches, and its focus about 40 inches ; of 
course it was of the polygonal construction familiar to 
lighthouse-keepers and their visitors, the grinding of 
a thirty-inch lens of continuous surface not having 
been contemplated in those days. Allowing for 
possible losses from surface-reflection or absorption by 
the glass, it was estimated that the lunar light and 
heat would be concentrated three thousand times. 

276 STUDY or the moon^s sui?face. 

One fine night in 1834_, near the time of fullmoon^ the 
lens and thermopile were put to the test. First the 
condensed beam of moon- rays was allowed to fall 
upon the pile and then it was screened loy an inter- 
posed board. The exposures and screenings were 
repeated many times ; but Professor Forbes was always 
disappointed with the effect, for it was nearly nil. 
There was a suspicion of movement in the galvano- 
meter needle, but the amplitude of the swing was 
microscapic, possibly not greater than a quarter of a 
degree. Assuming that this deflection may have re- 
sulted. Professor Forbes subsequently proceeded to 
estimate the amount of heat that it represented. By 
exposing his pile and a thermometer to one and the 
same source of artificial heat, he was enabled to insti- 
tute a comparison between the indications of each, and 
when he had done this and made all allowances for 
the condensing power of his lens, he concluded that 
the warming effect of the full moon upon our lower 
atmosphere was only equal to about the two hundred 
thousandth part of a centigrade degree ! 

"From what has sincebeenlearned, itappears strange 
that, with such a condensing power, such an insigni- 
ficant result should have come out ; but there was one 
point to which Forbes does not appear to have given 
the consideration it demanded. The sky was covered, 
he tells us, with a thin haze. Here was the secret, no 
doubt, of his comparative failure : this haze entirely 
cut off the little heat the moon had to give. When 
Melloni, using a similar lens, repeated his experiments 


under the pure sky of Naples,, lie saw his galvanometer 
swerve three or four degrees whenever the moon^s 
condensed light fell upon the pile ; from which he 
concluded that the moon gave warmth by no means 
insignificant^ though he did not take the pains to infer 
the actual degree upon any known scale. 

"This last essay of Melloni^s was made in 1846. Ten 
years elapsed before it was repeated, and then Pro- 
fessor Piazzi Smyth, who was about to test the advan- 
tages of a lofty astronomical station by carrying 
instruments to the summit of Teneriffe, placed this 
subject upon his programme, thinking reasonably, that 
in higher regions of the atmosphere -he might catch 
some of the warmth that is intercepted in its passage 
through these to the earth. He furnished himself 
with a pile and thermomultiplier, as the sensitive 
galvanometer has been termed ; but he used no lens, 
contenting himself with a polished metal cone in front 
of the pile to collect and reflect the lunar heat upon 
its face. There was no mistaking the effect at this 
elevation of 10,000 feet : when the cone was turned 
towards the moon, the needle swung towards the heat 
side of the scale through a perceptible angle, and 
when it was turned towards the sky opposite to the 
moon, the needle returned to zero. By repeating this 
alternation of exposures an average deflection was 
obtained which was free frojn the effects of slight dis- 
turbing causes. Then it became of interest to learn 
what this average deflection meant in terms of any 
terrestrial source of warmth, and Professor Smyth 


found that it was equivalent to one seventeenth part 
of that which his warm hand produced when it was 
held three feet from the pile, or about twice that of a 
Price's candle fifteen feet distant. He left as an after- 
work the conversion of this warmth into its equivalent 
on a known scale. The translation was quite recently 
made in France by M. Marie-Davy, and the result 
showed that the moon-heat experienced upon the 
mountain-top amounted to 750 millionths of a centi- 
grade degree. 

'^ "We come now to touch upon the recent more con- 
clusive experiments of the Earl of Rosse. When we 
look back upon the old trials, it is easy to see that the 
instruments employed, sensitive as they were, were 
yet not sufficiently so for the purpose. It seems that 
the want of delicacy was not in the thermopiles that 
converted the heat into weak electric currents, but in 
the galvanometers by which the weak currents were 
sought to be measured. Now these were formed of ordi- 
nary magnetic needles, poised upon points or turning 
upon pivots, the motion of the needle in each case 
being impeded to some extent by friction at its bear- 
ings. Then again, upon small, that is, short needles, 
feeble deflections are with difficulty seen, and those 
caused by the weak currents generated by moon-heat 
were, perhaps, too small to be seen at all. But it will 
be remembered that the requirements of sub-atlantic 
telegraphy brought about the invention of an exceed- 
ingly delicate galvanometer, in which the needle is 
suspended by a hair, and its most minute deflections 


are rendered visible by a small mirror which reflects 
a beam from an adjacent lamp on to a distant scale, 
so that an almost imperceptible twist of the needle 
causes a large displacement of the reflected light-spot. 
Here^ then, was an indicator capable of rendering 
visible the most feeble of electric currents generated 
in a thermopile. It was not invented long before it 
was turned to use by the astronomers. The Earl of 
Rosse was the first to test its capabilities upon the 

" Lord Rosse using a reflecting telescope of three 
feet aperture, set about measuring the lunar warmth, 
with a viev/ to estimating, first what proportion of it 
comes from the interior of the moon itself, and is not 
due to solar heating; second, that which falls from 
the sun upon the lunar surface, and is then reflected 
to us ; and third, that which falling from the sun upon 
the moon, is first absorbed by the latter and then ra- 
diated from it. We need not follow the instrumental 
details of the processes employed for the various deter- 
minations ; suffice it for us to know that the moon- 
heat was clearly felt, and that the quantity of warmth 
varied with the phase of the moon greatest at the 
time of full and least towards the period of new. From 
this it was evident that Httle or no heat pertains to the 
moon jper se ; that our satellite has no proper or in- 
ternal heat of its own, or at least that it does not 
radiate any such into space ; if it did, there would 
probably have been found evidence of a continuity of 
warming, independent of the -change of phase. Of the 


heat which came with the light only a small portion 
would pass through a glass screen in front of the pile; 
from this it was evident that the greater part of the 
whole consisted of heat-rays of low refrangibility ; 
from which Lord Rosse concludes that the major por- 
tion of the lunar warmth does consist of that solar 
heat which has first been absorbed by the moon and 
then radiated from it. 

'* By the aid of a vessel containing hot water, sub- 
tending the same angle at his pile as the reflector 
employed to condense the moon^s light and heat_, he 
was enabled to judge of the actual temperature which 
the lunar surface must have to produce the effect that 
it does ; and this was found to be about 500 degrees 
of Fahrenheit's scale. In this result we have a striking 
verification of a philosophical deduction reasoned out 
by Sir John Herschel, many years ago_, that ^ the sur- 
face of the full moon exposed to us must necessarily 
be very much heated, possibly to a degree much ex- 
ceeding that of boiling water.'* 

* These observations have recently been renewed under more 
favourable conditions. The result has been to show that a larger 
proportion of the moon's heat than had been supposed is reflected 
sun-heat. The difference in the radiation from the full moon and 
from the new indicates, according to these later observations, a 
difference of about 200 degrees in temperature. Moreover, during 
a partial eclipse of the moon on November 14, 1872, it was found 
that " the heat and light diminished nearly, if not quite, propor- 
tionally, the minimum for both occurring at or very near the middle 
of the eclipse, when they were reduced to about half their amounts- 
before and after contact with the penumbra." 


" Lord Rosse's conclusion that the heat increases 
with the extent of illumination has been confirmed 
by Marie-Davy, who has even measured the actual 
warmth day by day of a semi-lunation, and given 
the results in parts of the centigrade scale. He 
finds that the moon at first quarter warms the 
lower air by 17 millionths of a single degree, 
and that a regular increase takes place till about 
.the time of full moon, when the calorific effect 
reaches 94 millionths of a degree ! These insig- 
nificant figures refer only to the heat which can 
penetrate our atmosphere. The greater part of the 
whole lunar caloric must be absorbed in the high 
aerial regions. '^^ 

Here I must conclude my brief and necessarily im- 
perfect sketch of the researches which have been made 
into the aspect and condition of the moon-'s surface. 
Those who are desirous of extending their acquaintance 
with the subject should carefully study all the obser- 
vations which are recorded in the Proceedings of the 
Astronomical and Roy^l Societies, and the British 
Association, in this country, the leading Astronomical 
Societies on the Continent and in America, and the 
works in which Schroter, Gruithuisen, Madler, and 
Schmidt have dealt with lunar phenomena. But after 
all, no course of reading can prove so instructive or 
interesting as a thorough study of the moon^s surface 
with a telescope, even though the telescope be of 

* Fraser's Magazine for January, 1870. 


moderate power ; and I cannot better close tMs chapter 
than by earnestly recommending every student of 
astronomy to survey the lunar details as completely 
and systematically as his leisure and his instrumental 
means may permit. 




In discussing tlie nature of the celestial phenomena 
presented to lunarians, if such there be, we have con- 
siderations of two classes to deal with. In the first 
place, there are demonstrable facts respecting the 
apparent motions of the sun, earth, stars, and planets, 
the progress of the lunar seasons, year, and so on ; in 
the second, we have other points to consider re- 
specting which we can only form opinions more or 
less probable, as the possible existence of a lunar 
atmosphere of small extent, the nature and ejffects of 
such an atmosphere, the question whether life animal 
or vegetable exists on the moon, with other matters 
of a similar nature. 

But the only point of a doubtful nature respecting 
which I propose to speak at any length in this chapter, 
is the possible existence of a lunar atmosphere. All 
celestial phenomena must be so importantly afiected 
by the presence or absence of an atmosphere that it is 
desirable to inquire carefully into the evidence bearing 
on the subject. 

Remembering that our air is a mixture of oxygen 


and nitrogen (in the main), not a chemical compound 
of these gases, we see that there is no absolute neces- 
sity for the proportion in which these gases appear in 
our atmosphere. In the atmosphere of another body 
they might be differently proportioned. Moreover, 
carbonic acid gas, which forms a comparatively small 
part of the terrestrial atmosphere, might form a much 
larger proportion of the atmosphere of another planet. 
It is also conceivable that other and denser gases 
might be present in other atmospheres. 

But even when all such considerations as these have 
been taken into account, it remains certain that unless 
we assume the existence on the moon of gases un- 
known on earth, a lunar atmosphere would have a 
specific gravity, under like conditions of pressure, 
differing in no marked degree from that of our earth's 
atmosphere. It would be a somewhat bold assumption 
to take for the average specific gravity of the lunar 
atmosphere that of carbonic acid gas, which, as we 
know, is almost exactly half as great again as that of 
air. Bat even if we supposed the lunar atmosphere 
composed of a gas as heavy as chlorine (which has a 
specific gravity nearly 2| times as great as that of air), 
or like phosgene gas, which is nearly 3| times as 
heavy as air, the argument which follows would not 
be seriously affected. 

Our air is sufficient in quantity to form a layer 
about 5 1 miles in depth over the whole surface of the 
earth, and as dense throughout as air at the sea-level. 
This air, according 'to the laws of gaseous pressure. 


adjusts itself so that at any given height the density- 
corresponds to the quantity of air above that height. 
The air above any height acts, in fact, as a weight 
pressing upon the air at that height, and compressing 
its elastic substance until it has a density proportional 
to the pressure so produced. Obviously, therefore, 
the density of the air at any given level depends on the 
amount of the earth^s attraction. For every weight 
on the earth would be doubled if the earth's attraction 
were doubled, and halved if the earth's attraction 
were halved, and so on ; and this applies as fully to 
the air as to any other matter having weight. Ac- 
cordingly, if the earth's gravity were reduced to the 
value of gravity at the moon's surface (0*16 where the 
earth's gravity is represented by unity), the pressure 
of the air at the sea-level, and consequently the density 
of the air there, would be reduced to less than one- 
siKth of its present value. Of course, a given quantity 
of air at the sea-level would then occupy more space ; 
and the whole atmosphere would expand correspond- 
ingly. Instead of having to attain a height of about 
34 miles, as at present, before the pressure would be 
reduced to one-half that at the sea-level (or to y^^th that 
at present existing at the sea-level), it would be 
necessary to attain a height more than six times as 
great, or nearly 22 miles. In other words, instead of 
one half of the whole atmosphere lying as at present 
below the height of 8| miles, the lower half of the 
atmosphere would then extend to a height of nearly 
22 miles. 


Accordingly^ if on the moon there were an atmo- 
sphere constituted like ours, and sufficient in quantity 
to cover the moon's surface to a depth of about 5^ 
miles of uniform specific gravity equal to that of our 
air at the sea-level, then such an atmosphere under 
the moon's smaller attracting power would expand so 
greatly that the half nearest the moon would extend 
to a height of about 22 miles.* At the mean level of 
the moon's surface, that is, a level corresponding 
pretty nearly tcrour sea-level, so as to be as much 
above the greatest lunar depressions as below the 
greatest lunar heights, the pressure would be about 
one-sixth that at our sea-level. Thus it is seen that 
even though the lunarians had as much air per mile of 
surface as we have on the earth, they would have a 
much rarer atmosphere. At a height of seven miles 
from the earth, a greater height than has ever yet 
been attained, or than could be attained by man,t the 

* Here I take no account of the reduction of the moon's attracting 
power at this height from the surface. The consideration of such 
reduction would be important, however, in estimating the height 
to which the rarer strata would extend. 

t "In the celebrated ascent by Messrs. Glaisher and Coxwell, in 
which the greatest height yet reached by man was attained, Mr. 
Glaisher became insensible before the balloon had attained a height 
of six miles. Mr. Coxwell, after endeavouring to rouse Mr. Glaisher, 
found that he was himself losing his strength. Indeed, he was 
unable to use his hands, and had he not succeeded in pulling the 
valve-string with his teeth, he and his companion must inevitably 
have perished. The height attained before the string was pulled 
would seem, from an observation made by Mr. Coxwell, to have 
been about 6^ miles. At this time the temperature was 12 below 


air is still one-fourth as dense as at the sea-level. So 
that, even though the lunarians had so large a quantity 
of air as I have supposed, they must still be constituted 
very differently from men, since men would perish at 
once if placed in an atmosphere so attenuated. 

But there is a more important point to be consi- 
dered. "We see that an atmosphere of a given quan- 
tity per square mile of lunar surface would reach much 
higher than a similar atmosphere on the earth. One 
half of it would lie above a height of 22 miles, that is, 
enormously above the summits of the highest lunar 
mountains. Far the greater portion of the atmosphere 
would ile above the lunar high lands. Supposing the 
atmosphere differently constituted, and of specific 
gravity six times as great as our air under the same 

zero, and the neck of the balloon was covered with hoar frost.^' 
(From my article on the balloon in Rodwell's " Science Dictionary.") 
"It is worth noticing, however," I proceed, "that although it would 
seem from this experience that no man accustomed to breathe the 
air at ordinary levels, can hope to attain a greater height than 6| 
miles, it is not impossible that those who pass their lives at a great 
height, as the inhabitants of Potosi, Bogota, and Quito, might 
safely ascend to a far greater height. We know that De Saussure 
was unable to consult his instruments when he was at no higher 
level than these towns, and that even his guides fainted in trying 
to dig a small hole in the snow ; whereas the inhabitants of the 
towns thus exceptionally placed, are able to undergo violent exer- 
cise. We may assume, therefore, that their powers are exception- 
ally suited to such voyages as those in which Glaisher and Coxwell 
so nearly lost their lives." Nevertheless it may be regarded as 
certain that no race of men could exist even for a few minutes in 
an atmosphere having a specific gravity less than one-sixth that of 
our own air. 


circumstances of pressure^ yet even then we should 
have only the same density at the moon's level as at 
the earth's. That density could only be due to the 
pressure of the superincumbent parts of the atmo- 
sphere. Diminishing with height above the moon's 
mean surface, according to the laws of gaseous pres- 
sure, it would extend as high above the moon's surface 
as our air above the earth's,, even on the supposition 
of its having so remarkable a specific gravity compared 
with that of common air. 

We see, then, that if we were to suppose the atmo- 
spheric pressure at the moon's surface equal to that at 
the earth's, we should have to suppose either that this 
atmosphere is composed of gases of very great specific 
gravity, or else that it extends to a much greater 
height than our own atmosphere. In either case, it is 
obvious that we should expect to find very marked 
effects produced by such an atmosphere. 

In the first place, when the moon was carried by 
her motion over a star, the place of the star would be 
affected by refraction, not only when the moon's edge 
was very close to the star, but for some considerable 
time before. If the lunar atmosphere were actually 
as dense near the moon's mean surface as our air is at 
the sea-level, then a star would not be occulted at all, 
even though the moon passed so directly over the 
star's true place on the heavens that the geometrical 
line joining the star and the observer's eye passed 
through the moon's centre. This is easily seen. For the 


moon's semidiameter subtends an angle of less than 1 6\ 
Now the sun appears wholly in view when in reality- 
he is below the level of the horizon, our atmosphere 
having suflBcient refractive power to raise the sun's 
image by about 34' (his diameter is about ST). And 
this action is produced on rays which have only passed 
through the atmosphere to reach the earth tangen- 
tially. In passing out again, such rays would be 
-deflected through 34' more, or in all by about QS\ 
Accordingly, since 16' is less than a quarter of 68', if 
the moon's atmosphere possessed only a fourth part 
of the refractive power of our own atmosphere, a star 
in reality behind the centre of the moon's disc would 
appear as a ring of light. Nor would this ring be very 
faint. The light of the star would not be diluted or 
spread over the ring and therefore reduced in corre- 
sponding degree. On the contrary, the moon's atmo- 
sphere would act the part of an enormous lens, in- 
creasing the total quantity of light received from the 
star, in the same way that the lens of a telescope's 
object-glass increases the quantity of light received 
from any celestial object.--"^ 

" An efiect, indeed, somewhat similar to that here considered, 
may be produced by covering all but the outer ring of an object- 
glass with a black disc, and removing the eye-piece ; if then, the 
telescope be directed nearly towards a bright star, and shifted 
from that position until exactly directed on the star, the light 
from the star will be presented in the form of an arc, gradually 
extending farther and farther round until it forms a complete 
circular ring. 



In the case supposed^ as the moon really passed 
over a star^ we should see the star changing in 
appearance into an arc, this arc gradually increasing 
in length and span, until at length, when the star was 
centrally behind the moon, it would appear as a ring 
around her disc. 

The actual circumstances of an occultation of a star 
by the moon are very markedly contrasted with those 
here mentioned. In nearly all cases a star disappears 
instantly, when the moon's edge reaches the star's 
place. There is no perceptible displacement of the 
star, no change of colour, no effect whatever such as 
a refractive atmosphere would produce. In certain 
instances, the brightness of a star has been observed 
to diminish just before disappearance ; but we cannot 
be sure that, where this has happened, the star may 
not be really multiple, or perhaps nebulous. In the 
case of the star k Cancri, according to some ob- 
servers, the star has seemed suddenly reduced by 
about one half of its light, and almost instantly after 
to vanish ; but these phenomena, only noticed in the 
case of this star, may be fairly explained by sup- 
posing the star to be a close binary. Again, there 
have been instances where a star has seemed to 
advance for some distance upon the moon's disc 
before vanishing; but it is by no means unlikely 
that the star has in such a case chanced to cross 
the moon's limb where a valley or ravine has 
caused a notch or depression which is too small 
to be indicated by any ordinary method of observa- 


tion.* There is every reason to believe that when 
a single star is occulted opposite a smooth part of 
the moon^s limb^ the disappearance of the star is 
absolutely instantaneous. 

Moreover^ the evidence thus obtained has been 

" It is to be remembered that such disappearances as these 
always take place opposite the bright limb of the moon, for the 
dark limb, even when the moon is nearly new, cannot be properly 
seen. Accordingly, irradiation conies into play, as well, of course, 
as the ordinary optical diffraction of the images of points forming 
the lunar limb, both these causes tending to remove all trace of 
minute notches really existing on the limb. But when a star is 
occulted at such a notch, it of course remains visible, despite the 
irradiation of the moon's limb ; so that it seems to be shining through 
the moon's substance. That this explanation is sound, seems to be 
confirmed by the circumstance that observers at stations not very 
wide apart recognize different appearances. Take, for instance, 
the following passage from Smyth's " Celestial Cycle " : " One of 
the most remarkable projections of a star on the moon's disc which 
I ever observed, was that recorded in the fifth volume of the 
Astronomical Society's Memoirs, p. 363, of 119 Tauri, on the 18th 
of December, 1831. On that occasion the night was beautiful, the 
moon nearly full, and the telescope adjusted to the star which 
passed over the lunar disc, and did not disappear till it arrived 
between two protuberances on the moon's bright edge. This was 
also noted by Mr. Snow, p. 373 of the same volume ; but Sir James 
South saw nothing remarkable, although in a few minutes after- 
wards he observed the star 120 Tauri perform a similar feat." 
" Such anomalies," adds Smyth, " are truly singular." I cannot 
but think, however, that they are to be expected as a natural con- 
sequence of the unevennesses which certainly characterize certain 
parts of the lunar limb. Such unevennesses on the limb must be 
minute to escape detection through the effects of irradiation ; and 
accordingly a very slight difference in the position of two observers 
would suflSce to render the observed phenomena at their two 
stations altogether different. 

u 2 


strengthened by spectroscopic evidence. Dr. Huggins 
has watched the occultation of the spectrum of a 
star, that is to say, he has watched the spectrum of a 
star until the moment when the star itself has been 
occulted. He has found that the spectrum disappears 
as instantaneously as the star itself. Now this is well 
worth noticing; for it might be supposed that any 
atmosphere existing round the moon would affect the 
red rays more than the other ; as our atmosphere, for 
example, refracts the red light of the sun more fully 
than the rest. Hence it might be expected that 
the blue end of the spectrum would disappear a 
moment or two before the red end. But this did not 

The spectroscope has also afforded direct evidence 
of the non-existence of a lunar atmosphere of any 
considerable extent. For when the spectrum of the 
lunar light has been observed (by Dr. Huggins first, 
and later by others) it has been found to be absolutely 
similar to the solar spectrum, that is, there is no 
trace whatever of absorptive action exerted by a lunar 
atmosphere upon the solar rays which are reflected 
by her to the earth. This evidence is, of course, not 
demonstrative of the absolute want of air of any sort 
on the moon, because a very rare and shallow atmo- 
sphere would produce no appreciable absorptive effect ; 
but it confirms the other evidence showing that any 
lunar atmosphere must not only be extremely shallow 
but extremely rare. That is, there is not, as had been 
suggested by a well-known physicist, a dense atmo- 


sphere so shallow as not to rise above the summit of 
the lunar mountains. It is difficult, indeed, to conceive 
how such an atmosphere could be supposed to exist, 
since, as we have seen above, a gas six times as dense 
(under the same conditions) as our air, would on the 
moon only be as dense as our air, if so great in quan- 
tity as to reach as high as our air. An atmosphere 
sufficient in quantity to give traces of its presence in 
lunar shallows, but not extending higher than the 
summits of the lunar mountains, must be of a specific 
gravity so greatly exceeding (under the same condi- 
tions) that of common air, or indeed of any gas known 
to us on earth, that we are 'justified in regarding the 
theory of its existence as altogether unsupported by 

But perhaps the strongest evidence we have to 
show that the moon has either no atmosphere or so 
little that she may be regarded as practically airless, 
is to be found in the phenomena of solar eclipses. 
It is certain, in the first place, that if the moon had 
an atmosphere resembling the earth's, the sun would 
not disappear at all, even at the moment of central 
eclipse, and when the sun was at his smallest and the 
moon at her largest. The moon's atmosphere would 
act as a lens (or as part of a lens) and reveal the sun 
to our view as a ring of blazing lustre as really 
sunlight as the light of our setting sun. If the moon's 
atmosphere were at her mean surface but about one- 
fourth as dense as ours at the sea-level, the central 
part even of the sun's disc would be transmuted into 


a ring of light close to the moon^s edge, while the 
parts nearer the sun's edge would form outer and 
brighter parts of the ring of glory round the moon. 
A very shallow lunar atmosphere indeed would suffice 
to bring the parts close to the edge of the sun's disc 
into view. It was, indeed, once supposed that the 
sierra of red light seen round the moon's disc during 
total eclipse (that riug of red light which Mr. Lockyer 
so strangely supposed that he had discovered in 1868) 
was produced by the refraction of the sun's light by 
the moon.* We now know that no part of the 

* Thus Admiral Smyth wrote in 1844 : " The red flames or 
protuberances of light, observed during total eclipses, and so cor- 
rectly noted by the Astronomer Koyal and Mr. Baily during 
that of July 1842, seemed to be attributable to an atmospheric 
effect, albeit there may be no distinguishable atmosphere. So 
long ago as 1706, Captain Stannyan, at Berne, observed of the sun, 
' that his getting out of the eclipse was preceded by blood-red 
streaks of light from the left limb, which continued not longer 
than six or seven seconds of time.' On this Flamsteed remarks 
in a letter to the Royal Society : ' The Captain is the first man I 
ever heard of that took notice of a red streak of light preceding 
the emersion of the sun's body from a total eclipse ; and I take 
notice of it to you, because it infers that the moon has an atmo- 
sphere ; and its short continuation of only six or seven seconds of 
time tells us that its height is not more than the five- or six- 
hundredth part of her diameter.' This phenomenon was again 
noted during the total eclipse of the sun in April 1715, by Charles 
Hayes, the author of A treatise on Fluxions, who states in his 
philosophical dialogue Of the Moon that there was a streak of 
'dusky but strong red light' preceding the sun's reappearance. 
There is much uncertainty, however, in all these observations, 
from their being liable to so many conditions of place, weather, 
instrument, and wind." I quote the remainder of Admiral Smyth's 
remarks as bearing importantl}^ on our subject : " From more than 


light outside of the tnoon during totality is sunlight 
refracted by the moon, simply because the part where 

one observation, I had worked myself up to a belief that the globes 
of Saturn and Jupiter were more affected under occultation than 
could be assigned to the inflection of their hght in passing by the 
lunar surface ; and I also thought that I had seen the satellites 
of Jupiter change their figure at the instant of immersion. Thus 
prejudiced, so to say, I prepared to establish the point by the 
occultation of the 1st of June, 1831, and certainly observed it 
under a train of favouring circumstances ; but my result, as stated 
in the second volume of the Astronomical Society's Memoirs, p. 37, 
is this : Although the emersions of the satellites were perfectly 
distinct, they were certainly not so instantaneous as those of the 
small stars, which I think was owing more to light than disc. 
Jupiter entered into contact rather sluggishly ; but though the 
lunar limb was tremulous from haze, there was not the slightest 
loss of light. Faint scintillating rays preceded the emersion, which 
was so gradual, that, as the planet reappeared, the edge of the 
moon covered it with a perfectly even and black segment, which 
cut the belts distinctly, and formed clear sharp cusps, slowly 
altering until the whole body was clear. There was no appearance 
of raggedness from lunar mountains, and Jupiter's belts were 
superbly plain while emerging ; but there was not the slightest 
distortion of figure, diminution of light, or change of colour. . . . 
Schroter concluded that there existed a lunar atmosphere, but he 
estimated it to be only 5,742 feet high ; and Laplace considered it 
as being more attenuated than what is termed the vacuum in an 
air-pump. The slowness of the moon's motion on its axis may 
account for such result." (There is, however, no basis for this 
supposition.) ..." MM. Madler and Beer, whose selenographical 
researches have been carried to an unprecedented extent, arrive at 
the conclusion that the moon is not without an atmosphere, but 
that the smallness of her mass incapacitates her from holding an 
extensive covering of gas, and they add, ' it is possible that this 
weak envelope may sometimes, through local causes, in some 
measure dim or condense itself,' the which would explain some of 
the conflicting details of occultation phenomena." 


such refracted light would be strongest gives its own 
proper spectrum quite distinct from the spectrum of 
sunlight. But strong as this evidence is, there is yet 
stronger evidence. It has been discovered by Prof. 
Young that the sun has a relatively shallow atmo- 
sphere (say from two hundred to five hundred miles 
in height), whose existence is only rendered dis- 
cernible by spectroscopic analysis, aided by the moon. 
As the moon passes over the face of the sun, the 
visible sickle of the sun's disc grows narrower and 
narrower, until at last it vanishes; at that moment 
the shallow solar atmosphere is not yet covered, but 
is just about to be covered. For a moment or two 
the spectroscope gives the spectrum of this atmo- 
sphere, and this spectrum is found to consist of 
myriads of bright lines, the reversed Fraunhofer lines 
in fact. These are visible only for a second or two, 
and in the ordinary condition of the shallow atmo- 
sphere they vanish so suddenly that their disappear- 
ance has been compared to the vanishing of rocket 

* During the annular eclipse of June 1872, the lines were seen 
by Mr. Pogsou, Government Astronomer at Madras, for about 2 or 
3 seconds when the annulus was completed, and for about 6 or 
7 seconds when the annulus broke, showing a variable condition of 
the solar atmosphere. Moreover, the lines did not vanish suddenly 
in the latter case, as when the phenomenon was observed by Young 
in December 1870, and by Tennant, Maclear, and others, in Decem- 
ber 1871. These peculiarities have no bearing on the question of 
the moon's atmosphere, but I thought it desirable to mention them, 
lest the reader should derive erroneous impressions from the account 
given above. The general subject of the sun's complex shallow 


Now if tlie moon had an atmosphere comparable 
even with what is called the vacuum of an air-pump, 
the recognition of the delicate phenomena attesting 
the existence of the shallow solar atmosphere would be 
wholly impossible. The slightest residue of sunlight 
brought into action by the refractive power of such 
an atmosphere would suffice to obliterate the beauti- 
ful but delicate spectrum of the complex solar atmo- 
spheric envelope. 

The evidence derived from the non-existence of 
any twilight circle on the moon, or the extreme 
narrowness of any such zone which may exist, need not 
here be closely considered. The only observations 
yet made which appear to indicate the existence of 
lunar twilight, seem explicable as due to the fact that 
the sun is not a point of light illuminating the moon's 
surface, but presents, as seen from the moon, a disc 
as large as she shows to us. Thus there would be in 
the case of a smooth moon, a penumbral fringe border- 
ing the illuminated hemisphere, aud about 32' of the 
arc of a lunar great circle in width. This would 
correspond to a breadth of nearly ten miles, and would 
be readily discernible from the earth. In the case of 
a rough body like the moon, there would be no regular 
penumbral fringe, but along some parts of the border- 
line between light and darkness the effect would be 

atmosphere is fully discussed in iriy treatise on ' The Sun,' in the 
first edition of which I adopted the theory that such an atmosphere 
must exist, while as yet the decisive observations remained to be 


reduced^ while along other parts it would be ex- 
aggerated. On the whole, there would result appear- 
ances closely resembling those due to a twilight circle 
of small extent ; and we can reasonably ascribe the 
supposed twilight effects hitherto recognized to this 
cause, that is, to the fact that the sun as seen from 
the moon is not a point of light but a disc. 

The conclusion to which we seem forced by all the 
evidence obtainable, is that either the moon has no 
atmosphere at all (which scarcely seems possible), or 
that her atmosphere is of such extreme tenuity as not 
to be perceptible by any means of observation we can 
apply. I must, however, make some remarks here 
on a theory which has been advocated by astronomers 
of repute, and even discussed by Sir John Herschel 
as not wholly incredible, the theory, namely, that a 
lunar atmosphere (and lunar oceans) may possibly 
exist on the hemisphere of the moon which is turned 
directly away from the earth. 

This theory is based on another, the theory, namely, 
that the moon's centre of gravity is nearer to us than 
her centre of ligure. Thus Professor Hansen considers 
that an observed discrepancy between the actual lunar 
motions and the results of the theoretical examination of 
the moon's inequalities, is removed if the centre of 
gravity of the moon is assumed to be 33^ miles farther 
from the earth than her centre of figure. This result 
which, however. Professor Newcomb questions 
appears to have been confirmed by the comparison of 
photographic pictures of the moon, taken at the times 


of her extreme eastern and western librations. In 
the year 1862, M. Gussew, Director of the Imperial 
Observatory at Wilna, carefully examined two such 
pictures taken by Dr. De la Rue. The result of the 
examination may be thus stated : The outer parts of 
the visible lunar disc belong to a sphere having a 
radius of 1,082 miles, the central parts to a sphere 
having a radius of 1,063 miles; the centre of the 
smaller sphere is about 79 miles nearer to us than 
the centre of the larger ; the line joining the centres 
is inclined at an angle of about 5 to the line from 
the earth at the epoch of mean libration : thus the 
central part of the moon's disc is about 60 miles 
nearer to us than it would be if the moon were a 
sphere of the dimensions indicated by the disc's out- 
line. If we suppose the invisible part of the moon's 
surface to belong to the larger sphere, and the density 
of the moon's substance uniform, it would follow from 
this conformation that the centre of gravity of the 
moon is about 30 miles farther from the earth than 
is the middle point of the lunar diameter directed 
towards the earth, that is, than is the centre of the 
moon's apparent figure. This result accords sufficiently 
well with Hansen's theoretical conclusion. 

On this Sir John Herschel remarks : " Let us now 
consider what may be expected to be the distribution 
of air, water, or other fluid on the surface of such a 
globe, supposing its quantity not sufficient to cover 
and drown the whole mass. It will run towards the 
lowest place, that is to say, not the nearest to the 


centre of figure, or to the central point of tlie mere 
space occupied by the moon, but to the centre of the 
mass, or the centre of gravity. There will be formed 
there an ocean of more or less extent, according to the 
quantity of fluid directly over the heavier nucleus, 
while the lighter portion of the solid material will 

stand out as a continent on the opposite side In 

what regards its assumption of a definite level, air 
obeys precisely the same hydrostatical laws as water. 
The lunar atmosphere would rest upon the lunar 
ocean, and form in its basin a lahe of air, whose upper 
portions, at an altitude such as we are now con- 
templating, would be of excessive tenuity, especially 
should the lunar provision of air be less abundant 
in proportion than our own. It by no means follows, 
then, from the absence of visible indications of water 
or air on this side of the moon, that the other is 
equally destitute of them, and equally unfitted for 
maintaining animal or vegetable life. Some slight 
approach to such a state of things actually obtains on 
the earth itself. Nearly all the land is collected in one 
of its hemispheres, and much the larger portion of 
the sea in the opposite. There is evidently an excess 
of heavy material vertically beneath the middle of the 
Pacific ; while not very remote from the point of the 
globe diametrically opposite rises the great table-land 
of India and the Himalaya chain, on the summits of 
which the air has not more than a third of the density 
it has on the sea-level, and from which animated 
existence is for ever excluded. '^ 


But pleasing thougli the idea may be that on the 
farther hemisphere of the moon there may be oceans 
and an atmosphere^ it appears to me impossible to 
accept this theory. In the first place, it has not been 
demonstrated, and is in fact not in accordance with 
theoretical considerations, that the moon is egg- 
shaped, or bispherical, according to Gussew's view. 
The farther part may also project as the nearer part 
does (supposing Gussew^s measurements and in- 
ferences to be trustworthy). But even if we assume 
the moon to have the figure assigned to it by Gussew, 
the invisible part is not that towards w^hich the 
atmosphere would tend. The part of the surface 
opposite the centre of the visible disc is in fact not 
nearest to the centre of gravity, but (assuming the 
unseen part spherical, and of the radius indicated by 
the visible disc) is 30 miles farther from the centre of 
gravity than are points on the edge of the visible disc. 
The band or zone of the moon's surface lying on this 
edge is the region where oceans and an atmosphere 
should be collected* (if water and air existed in 
appreciable quantity) on the moon's surface. 

* The argument is presented in another form in a paper con- 
tributed by me to the Monthly Notices of the Astronomical Society, 
as follows : " Let us assume, with Hansen, that the moon's surface 
is formed of two spherical surfaces, the part nearest to us having 
the least radius, so that in fact the moon is shaped like a sphere to 
which a meniscus is added, said meniscus lying on the visible hemi- 
sphere. If we imagine the meniscus removed, the lunar atmosphere 
would dispose itself symmetrically round the moon's spherical sur- 


We seem justified in considering the phenomena 
presented to an observer supposed to be stationed on 
the mooUj as practically those which would be seen if 
the moon had no atmosphere at all. 

face. Now, suppose that while this state of things exists, the lunar 
air within the region now occupied by the meniscus of solid matter 
is suddenly changed to matter of the moon's mean density, what 
could be the effect of this change, by which new matter would be 
added on the side of the moon towards the earth ? Surely not that 
the remaining atmosphere would tend to the further side of the 
moon, but on the contrary that it would be attracted towards the 
nearer side by the new matter there added. The lunar air would 
be shallower on this nearer side, no doubt, because the air thus 
drawn to it would not make up for the air supposed to be changed 
into the solid form ; but at the parts which form the edge of the 
disc there would be an access of air, without this diminishing cause, 
and the air would therefore be denser there than elsewhere. But 
in this final state of things there would be equilibrium ; we learn 
then what are the conditions of equilibrium for a lunar atmosphere, 
assuming the moon's globe to have the figure supposed by Hansen. 
There would be a shallow region in the middle of the visible disc, 
and a region slightly shallow directly opposite, while the mid-zone 
would have the deepest atmosphere. But it is around this zone 
precisely that no signs of a lunar atmosphere have as yet been 
recognized. I may remark that this reasoning may be extended 
to the earth. Assuming the waters of the earth drawn towards 
the South Pole because of a displacement in the earth's centre of 
gravity, we may regard the surface of the sea in the southern 
hemisphere as standing above the mean surface of the globe, and 
a part of the southern seas as therefore constituting a meniscus 
like that conceived by Hansen to exist in the case of the moon. 
It would follow, then, if my reasoning be correct, that we should 
have the atmosphere shallowest in high southern latitudes shallow, 
but only slightly so, in high northern latitudes, and densest between 
the tropics ; but this, as is well known, is precisely the observed 


These phenomena may be divided into celestial and 

Of lunarian phenomena_, that is^ of the appearance 
presented by lunar landscapes^ I shall say little ; be- 
cause^ in point of fact_, we know far too little respecting* 
the real details of lunar scenery to form any satisfactory 
opinion on the subject. If a landscape-painter were 
invited to draw a picture presenting his conceptions 
of the scenery of a region which he had only viewed 
from a distance of a hundred miles, he would be under 
no greater difficulties than the astronomer who under- 
takes to draw a lunar landscape, as it would actually 
appear to any one placed on the surface of the moon. 
We know certain facts, we know that there are 
striking forms of irregularity, that the shadows must 
be much darker as well during the lunar day as during 
an earthlit lunar night, than on our own earth in 
sunlight or moonlight, and we know that whatever 
features of our own landscapes are certainly due to the 
action of water in river, rain, or flood, to the action of 
wind and weather, or to the growth of forms of 
vegetation with which we are familiar, ought assuredly 
not to be shown in any lunar landscape. But a 
multitude of details absolutely necessary for the due 
presentation of lunar scenery are absolutely unknown 
to us. Nor is it so easy as many imagine to draw 
a landscape which shall be correct even as respects 
the circumstances known to us. For instance, though 
I have seen many pictures called lunar landscapes, I 
have never seen one in which there have not been 


features manifestly due to weathering and to the action 
of running water. The shadows again are never shown 
as they would be actually seen if regions of the in- 
dicated configuration were illuminated by a sun but 
not by a sky of light. Again^ aerial perspective is 
never totally abandoned, as it ought to be in any 
delineation of lunar scenery. 

I do not profess to have done better, myself, in the 
so-called lunar landscapes which illustrate this chapter. 
I have, in fact, cared rather to indicate the celestial 
than the lunarian features shown in these drawings. 
Still, I have selected a class of lunar objects which 
may be regarded as on the whole more characteristic 
than the mountain scenery usually exhibited. And 
by picturing the greater part of the landscape as at a 
considerable distance, I have been freer to reproduce 
what the telescope actually reveals. In looking at 
one of these views, the observer must suppose himself 
stationed at the summit of some very lofty peak, and 
that the view shows only a very small portion of what 
would really be seen under such circumstances in any 
particular direction. The portion of the sky shown in 
either picture extends only a few degrees from the 
horizon, as is manifest from the dimensions of the 
earth's disc ; and thus it is shown that only a few 
degrees of the horizon are included in the landscape. 

Now, as respects the celestial phenomena visible 
from the moon's surface, much more might be said. 
The subject would afford a mathematician who had 
sufficient leisure a fine field for very unprofitable 


labour. He could determine the exact course of every 
star in the sky of every celestial hemisphere for every 
latitude in the moon. He could discuss the seeming 
motions of the sun with great completeness, and lastly 
he could indicate the exact nature of the varying curves 
traced out by the centre of the earth's disc as it librated 
responsively to the moon's varying motions in her 
orbit, until he had shown that the minutest detail of 
the lunar theory as already mastered, as well as yet 
minuter details of the lunar theory of the future, have 
their analogues in the apparent motions of the earth 
as seen from the moon. All this he might do, and 
the work, to be properly achieved, would try to their 
utmost the powers of a Newton. But when the work 
was finished it would be of very little worth ; a new 
Diamond might set the new Newton's manuscript in 
flames without deserving even the mild reproach'^ Oh, 
Diamond, Diamond ! thou little knowest the mischief 
thou hast done." 

These matters, then, must here be more summarily 
disposed of. 

As to the stars, we have these general considera- 
tions : 

First, the stars must be visible day and night, since 
the lunar sky in the daytime must be perfectly black, 
except where the sun's corona and the zodiacal light 
spread a faint light over it ; and even where this light 
is, the stars must be quite clearly visible. Secondly, 
many orders of stars below the faintest discernible by 
our vision must be visible in the lunar heavens to 



eyesight such as onrs_, by day as well as by night. 
The Milky Way^ in particular, must present a magni- 
ficent spectacle.* 

The apparent motions of the stars correspond to the 
moon^s rotation. Since she turns on her axis once in 
27'322 of our days, and in the same direction that our 
earth turns, it follows that the star-sphere turns round 
from east to west as with us, but at a rate more than 
twenty-seven times slower. The pole of the lunar 
heavens lies close to the pole of the ecliptic, since the 
inclination of the moon's axis is only 1| degrees. But 
the pole shifts more quickly than the pole of our 
heavens, completing its circuit around the pole of the 
ecliptic, in a circle 3 degrees in diameter, in 18*6 
years. Thus in the course of a lunar day the pole of 
the heavens shifts appreciably in position, and there- 
fore the stars do not travel in true circles, nor remain 
at a constant distance from the pole of the heavens 
(as our stars appreciably do). This noticed, the 
motion of the star- sphere, except as to rate, corre- 
sponds latitude for latitude with that of our star-sphere. 
The northern and southern poles of the heavens are 
overhead to observers placed respectively at the 
northern and southern poles of the moon ; and as the 
lunarian travels towards the equator from a northern 
or southern station, the pole descends along a northerly 

* I have not ventured to include any part of the Milky Way in 
the pictures illustrating this chapter, for this reason simply, that 
no ordinary engraving could give the slightest idea of the splendour 
of the galaxy as seen from an airless planet. 


or southerly meridian respectively^ until at the lunar 
equator the two poles are both on the horizon. The 
equator of the lunar star-sphere lies always close to 
the ecHptic^ the points corresponding to those parts 
of our celestial equator which lie farthest from the 
ecliptic_, being only 1| degrees instead of about 23^ 
degrees, as with us, from the ecliptic. These points 
and the nodes of the equator shift round so as to 
perform a complete circuit of the ecliptic in 18' 6 years. 

The motions of the sun bear the same relation to 
the star-motions as in the case of our own celestial 
phenomena. As our solar day exceeds the sidereal 
day, on account of the sun^s advance on the ecliptic, 
so the solar day on the moon exceeds the sidereal 
lunar day, amounting to 29*531 of our terrestrial days, 
instead of 27"o22 days. 

But while the lunar day is much longer than ours, 
the lunar year is considerably shorter. For the 
precession of the nodes is, as we have seen, much 
more rapid in the moon^s case than in the earth's ; 
and the lunar tropical year, which is of course the 
year of seasons, is correspondingly shortened by the 
rapid motion of the vernal and autumnal equinox- 
points to meet the sun as he advances along the 
ecliptic. We know precisely what the lunar tropical 
year is, from the result stated at p. 134. It lasts 
346*607 of our days, or 11*737 lunar days. Thus on 
the average each lunar season spring, summer, 
autumn, or winter lasts 2*934 lunar days, or nearly 
three days. But the seasons are not very marked, 
X 2 


since the sun^s range is only from 1^ north to 1^ 
south of the ecliptic, which is rather less than the 
range of our sun during four days before and after 
either equinox, vernal or autumnal. It appears to me 
that this state of things scarcely warrants the state- 
ment of Sir W. Herschel that " the moon^s situation 
with respect to the sun is much like that of the earth, 
and by a rotation on its axis it enjoys an agreeable 
variety of seasons and of day and night.^^ 

Differences of climate exist, however, on the moon ; 
and the circumstance is one to be carefully borne in 
mind in discussing the physical condition of our satel- 
lite. Day and night are nearly equal everywhere on 
the moon^s surface, and during all the year of twelve 
long days. Moreover, the sun everywhere and at all 
times rises nearly due east and sets nearly due west. 
But his meridian altitude varies with latitude precisely 
as the meridian altitude of our spring or autumn sun 
varies with latitude. Along the lunar equator he rises 
to the point overhead, or very near to it, at mid-day ; 
and the same may be said of all places within the 
lunar tropical zone (three degrees only in width). 
Near the lunar poles, on the contrary, the mid-day 
sun is close to the horizon. And in mid latitudes the 
mid-day sun has an intermediate altitude, which is 
greater or less according as the place is nearer or 
farther from the equator. 

The motions of the planets as seen from the moon 
need not be fully discussed. It may be noted that all 
the motions of advance and retrogression observed 


from the earth can be seen from the moon also. The 
principal difference in the view of the planets obtained 
from the lunar station consists first in the visibility of 
A'^enus and Mercury when close to the sun, so that the 
varying illumination of these planets can be traced 
during their complete circuit around the sun, and 
secondly, in the visibility not only of Uranus, Nep- 
tune, and whatever other planets may travel beyond 
Neptune, but of many hundreds, and perhaps thou- 
sands, of the asteroids. If any planet or planets 
travel within the orbit of Mercury, lunarian astrono- 
mers, if such there be, must bo well aware of the fact, 
supposing their powers of vision equal to ours. 

The solar surroundings, as the prominences, corona, 
zodiacal, meteor systems, comet-families, and so on, 
must be perfectly visible from the moon ; and in par- 
ticular, before sunrise and after sunset these objects 
must form a very striking feature of the lunar heavens. 
I shall presently venture to give a brief ideal sketch 
of some of the more remarkable circumstances of the 
scene presented to a supposed lunar observer, as these 
and other phenomena pass in review before him. The 
reader will find in this sketch a description of the 
probable appearances presented during an eclipse of 
the sun by the earth. 

But it is in the phenomena presented by our earth 
herself that our imagined lunarians must find their 
most interesting and difficult subject of study. On 
her they have an object of contemplation utterly 
unlike anv known to our astronomers. 



Of course on the farther side of the moon, at least 
on those parts which are never brought into view by 
libration^ the lunarians never see the earth at all. 
On the hither side she is at all times visible^ though 
under very varying conditions of illumination. On 
the zone including places on the moon which alter- 
nately pass into view and out of view, she is alternately 
seen and concealed, but to varying degrees. 

Thus let us begin with the ^ parallactic fringe ' 
pmein'j)^, next to the illuminated region, shown in 
fig. 79, Plate XVI. The inner edge of this fringe 
(the left-hand edge in the figure) indicates a line on 
the moon where the earth's centre in extreme lib ra- 
tions just descends to the horizon, but never below the 
horizon. The outer edge marks a line where the 
earth's whole disc just disappears in extreme libra- 
tions. Thus on places within this fringe-region the 
earth sometimes descends so low as to show less than 
half her disc above the horizon. If a fringe equally 
wide were drawn just within the inner edge of this 
fringe, it would include all places where the earth in 
extreme librations descends so low that some part 
of her disc (less than half) is concealed below the 
horizon . 

Next let us take the space marked as the ' Kegion 
carried out of view by libration.' This is the lunar 
zone where the earth passes out of view in libration, 
but is for the greater part of the time in view, wholly 
or partially. The next zone, marked as the ' Region 
brought into view by libration,' is the lunar zone 


where the earth passes into view in libration^ but is 
for the greater part of the time wholly or partially 

Lastly, let us take the parallactic fringe p m e m' p'. 
Here the edge next to the unseen region indicates a 
line on the moon where, in extreme librations, the 
earth^s edge just touches the horizon, no part of the 
earth becoming visible. The other edge indicates a 
line on the moon where, in extreme librations, the 
earth^s centre just reaches the liorizon. On any place, 
therefore, within the fringe, a part of the eartVs disc, 
but always less than a half, becomes visible in extreme 
librations. A fringe as wide as the other side of the 
line p m e m' p', includes places on the moon where, 
in extreme librations, more than the half of the earth^s 
disc becomes visible, but not the whole disc, and its 
right-hand edge is a line on the moon where, in 
extreme librations, the whole disc of the earthjust 
becomes visible, touching the horizon at its lower 

Now as to the actual motions of the earth's disc as 
seen either at places on these several zones, or at any 
part of the region where the earth is always above the 
lunar horizon, we have, as I have mentioned above, 
a problem which, in its entirety, is of the most com- 
plicated kind. But all that is useful to be known 
can be inferred from a few simple considerations, which 
I proceed now to indicate. 

In fig. 72, Plate XV., the moon's disc is supposed 
to be indicated, and the range of libration at thirteen 


points on that disc, the actual libration-curves being 
illustrated in figs. 78, 74, 75, and 76. Now at a point 
close to P, the north pole of the moon, and on P P', 
the earth is seen close to the south point of the horizon. 
(This is manifest, if we consider that a line drawn 
from the eye to a point near P, is nearly tangential to 
the half-sphere PP', and extends from near P in a 
direction which, estimated by a lunarian, would lie on 
the meridian P P'.) Again, at 0, the earth must be 
seen overhead. And it is clear that, to a lunarian 
travelling uniformly from P to 0, and thence to P', 
the earth would be seen to pass from the southern 
horizon * to the point overhead, and thence to the 
northern horizon, by an equally uniform motion. 

* Before our lunarian left the point P he would have no deter- 
mined southern point on the horizon. In fact, to a person at the 
pole of a globe like the moon or earth, every point of the horizon 
is towards the south ; as soon as he moves from the pole, be it 
only by a single step, the point of the horizon towards which he 
has thus moved becomes the south point ; and then, of course, 
the point behind him is the north point, the point on his right is 
the west point, and the point on his left the east point. 

It seems perplexing to many, viewing the problem dealt with 
above, that a lunarian near P seeing the earth towards the south, 
while a lunarian at E sees the earth towards the east, a lunarian at 
Mo should see the earth towards the east, and not towards the 
south-east. But this depends on precisely the same considerations 
which show how the sun, in spring or autumn, rises due east for 
all parts of the earth. It will be observed that the geometrical 
explanation consists in the fact, that if we draw lines M^ M4, 
M2 Ml in fig. 72, and regard these as the orthogonal projections 
of circles, both these circles cross the arc P E at M at right angles. 
It should be noted, that whereas we call E the eastern and E' the 


Again, suppose a lunarian to travel I'rom Mj to M^ 
on a lunar latitude-j)arallel (which seen from the earth 
would appear as a straight line from Mo to M^) . It is 
manifest that as he thus travelled, the earth would 
change in apparent position precisely as though he 
were being carried round by a rotation of the moon on 
the axis P P'_, which would necessarily produce the 
same effects as we on earth recognize in the sun's 
daily motion. The earth, to our travelling lunarian, 
would seem to pass from the eastern horizon to the 
south, where her elevation (from what was shown in 
the preceding paragraph) would be equal to the arc 
P Mg or P Ml. 

Accordingly, the apparent place of the earth, as 
seen from any point of a latitude-parallel Mj M^, on 
the moon's disc, is at once determinable by the con- 
sideration that the earth lies on the half-circle from the 
eastern to the western horizon of the given place on 
the moon, this half-circle being inclined to the horizon 
by as many degrees as there are in the arc PMo, and 
the earth being as many degrees (measured along 
this circle) from the eastern point as the given point 
on the moon is from M2 (these degrees being measured 
along the latitude-parallel Mg MJ. 

We can thus at once determine on what part of the 
lunar sky the earth is seen from any given place on 
the moon. Thus taking the intersections of the cross 

western edge of the moon's disc (because so seen on the sky), a 
lunarian at Avould have his east towards E' and his west 
towards E. 


lines in fig. 72_, as conveniently and sufiiciently illus- 
trating the above reasonings we liave^ as the parts of 
the skv where the earth is seen, the following results, 
placed so as to correspond to the position of the thirteen 
intersection-points in fig. 72 : 


S. 30 

Abt. S.E. 23-^ S. 60= S.W. 23 abt. 

E. E. 30 E. 60 Overhead. W. 60 W. 30 W. 

Abt. N.E. 25 N. 60 N.W. 23 abt. 

N. 30 

Having thus ascertained the mean position of the 
earth on the lunar sky for any given lunar station, we 
can infer the nature of the libratory motions of the 
earth about this mean position. We see that the 
libration in longitude, if it acted alone, would neces- 
sarily sway the earth backwards and forwards along 
an arc of about 14f degrees (that is 7~ degrees on 
either side of her mean position) upon the circle 
through the east and west horizon-points. This is 
obvious, because the libration in longitude necessarily 


takes place along a lunar latitude-parallel. We can 
also readily determine the position of the line on the 
heavens along which the earth is shifted by the 
libration in longitude in any given case. Thus, take 
the libration-cross near Mg, fig. 72, Plate XY. We 
see that its upper end, corresponding to the time when 
the earth is lowest down, is separated by a smaller 
latitude-arc from the eastern edge of the disc than the 
mean libration-point. This shows that the earth, in 
leaving her mean position southwards, or descendingly, 
shifts also eastwards ; and, of course, in leaving her 
mean position northwards, she shifts westwards. Thus 
we have a libration-arc inclined at an acute angle to 
the libration-arc before considered. The corresponding 
motions of the earth^s centre are therefore not like 
fig. 76, but as this figure would appear if D W were 
inclined to d d' at an acute angle. 

It would be idle, however, to enter into further 
details on these points, simply because the result would 
have no value. It is indeed instructive to consider 
the general features of the heavens as seen from any 
celestial body, and the general fact that the earth, 
as seen from each lunar station on the visible hemi- 
sphere, has such and such a mean position, and sways 
libratingly around or across that position, is sufiiciently 
interesting. But the special circumstances of these 
librations have no interest, because in no sense affect- 
ing the physical habitudes of the different lunar regions. 
Moreover, a volume much larger than the present 
would be required for their adequate discussion. 


It is manifest that at each lunar station the earth 
changes in phase precisely as the moon changes with 
us. When we see the moon full, the lunarians have 
the earth ' new/ that is, wholly dark ; when we see 
the moon at her third quarter, the earth, as seen from 
the moon, is at her first quarter ; when the moon is 
new, the earth is ' full ' ; and, lastly, when the moon is 
at her first quarter, the earth is at her third quarter. 
But in the case of the earth seen from the moon, the 
changes are all gone through while she is in one and 
the same part of the heavens ; and though they neces- 
sarily depend on the sun's distance from the earth, 
this distance changes by the sun's apparent motion 
around the lunar heavens, and not, as in the case of the 
moon, by the motions chiefly of the lesser luminary. 
Moreover it is manifest that the earth's phases occur 
at different hours of the lunar day at different sta- 
tions. Where the earth is seen on the meridian, ' new 
earth ' necessarily occurs at noon-day, ' first quarter ' 
at sunset, ' full earth' at midnight, and ^ third quarter ' 
at sunrise. Where the earth is seen on the east of the 
meridian, ' new earth ' occurs in the forenoon, ' first 
quarter ' in the afternoon, ' full earth ' between sunset 
and midnight, and ' third quarter ' between midnight 
and sunrise. Where the earth is seen on the west of 
the meridian, ^ new earth ' occurs in the afternoon, 
' first quarter ' between sunset and midnight, ' full 
earth' between midnight and sunrise, and third 
quarter in the forenoon. 

Again, the earth changes in aspect to the lunarians 


on account of the inclination of her axis. When the 
moon is north of the equator, the lunarians see the 
north polar regions, or have a view of the earth 
resembling a summer sun-view of the earth;* when 
the moon is south of the equator, the lunarians see the 
south polar regions, or a view resembling a winter 
sun-view of the earth. These changes correspond ex- 
actly, in sequence, with the varying sun-views of the 
earth during a year, since the moon, like the sun, 
passes alternately north and south of the equator 
as she travels towards the east on the heavens. But 
the period of these changes, in the case of the moon, 
is of course the period occupied by the moon in passing 
from the equator to her greatest northerly declination, 
thence to the equator, again to her greatest southerly 
declination, and finally to the equator once more, and 
this period has a mean value equal to a nodical month. 
It will be manifest from fig. 54, Plate XIII., and the 
explanation in pp. 158 164, that the range of the 
earth's apparent sway, by which her north and south 
poles are brought alternately into view, varies from 
18 18' to 28 35' on either side of the mean position 
(when both poles are on the edge of her visible disc). 
The period in which these changes are completely 
passed through is of course that of the revolution of 
the moon's nodes, or 18*6 years. 

In the considerations here dealt with, the student 
who has sufficient leisure will find the necessary 

* See my " Sunviews of the Earth." 


materials for the complete discussion of tlie varying 
aspect and position of the earth as supposed to be 
seen from any lunar station. 

Before drawing this chapter to a conclusion^ however^ 
I shall venture to attempt the description of some of 
the chief events of a lunar month, as they might be 
supposed actually to present themselves if an inhabi- 
tant of earth could visit the moon and observe them 
for himself. I select time and place so as to include 
in the description the phenomena of an eclipse of the 
sun by the earth. The reader will perceive that 
neither of the views illustrating this chapter cor- 
responds with the relations considered in the following 
paragraphs, in fact, it was absolutely necessary to 
select for pictorial illustration a lunar station where 
the earth would be low down, whereas for descriptive 
illustration it was manifestly better to take a station 
having the earth high above the lunar horizon. 

To an observer stationed upon a summit of the 
lunar Apennines on the evening of November 1, 
1872, a scene was presented unlike any known to the 
inhabitants of earth. It was near the middle of 
the long lunar night. On a sky of inky blackness 
stars innumerable were spread, amongst which the 
orbs forming our constellations could be recognized 
by their superior lustre, but yet were almost lost 
amidst myriads of stars unseen by the inhabitants of 
earth. Nearly overhead shone the Pleiads, closely 
girt round by hundreds of lesser Hghts. From them 
towards Aldebaran and the clustering Hyads, and 


onwards to the belted Orion, streams and convolutions 
of stars, interwoven as in fantastic garlands, marked 
the presence of that mysterious branch-like extension 
of the Milky Way which the observer on earth can 
with unaided vision tx-ace no farther than the winged 
foot of Perseus. High overhead, and towards the 
north, the Milky Way shone resplendent, like a vast 
inclined arch, full " thick inlaid with patines of bright 
.gold/' Instead of that faint cloud-like zone known 
to terrestrial astronomers, the galaxy presented itself 
as an infinitely complicated star region, 

" With isles of hght and silvery streams, 
And gloomy griefs of mystic shade." 

On all sides, this mighty star-belt spread its out- 
lying bands of stars, far away on the one hand towards 
Lyra and Bootes, where on earth we see no traces of 
milky lustre, and on the other towards the Twins and 
the clustering glories of Cancer, the ^dark con- 
stellation ^ of the ancients, but full of telescopic splen- 
dours. Most marvellous too appeared the great dark 
gap which lies between the Milky Way and Taurus . 
here, in the very heart of the richest region of the 
heavens, with Orion and the Hyades and Pleiades 
blazing on one side, and on the other the splendid 
stream laving the feet of the Twins, there lay a deep 
black gulf which seemed like an opening through our 
star system into starless depths beyond. 

Yet, though the sky was thus aglow with star-light, 
though stars far fainter than the least we see on the 


clearest and darkest night were shining in countless 
myriads, an orb was above the horizon whose light 
would pale the lustre of our brightest stars. This 
orb occupied a space on the heavens more than twelve 
times larger than is occupied by the full moon as we 
see her. Its light, unlike the moon's, was tinted with 
beautiful and well-marked colours. At the border, 
th& light of this globe was white, while somewhat to 
the left of the uppermost point, and as much to the 
right of the lowest, a white light of peculiar purity 
and brilliancy extended for some distance upon the 
disc. But whereas the upper passed farthest round 
the disc's edge, and seemed on the whole to be the 
most extensive, the lower spread farther in upon 
the disc, and appeared rounded into an oval shape. 
Corresponding to this peculiarity was the circumstance 
that the greater part of the disc's upper half was 
occupied by a misty and generally whitish light, 
amidst which spots of blue could be seen on the right 
and left, and brownish and yellowish streaks near the 
middle ; while, on the contrary, the lower half of the 
disc was nearly free from misty light, and occupied on 
the sides by widely- extended blue regions, and in the 
middle by green tracts on a somewhat yellowish 
background. To an inhabitant of earth it would not 
have been difficult to recognize in this last-named 
region the continent of South America bathed in the 
full light of a southern summer sun. 

The globe which thus adorned the lunar sky and 
illuminated the lunar lands with a light far exceeding 


tliat of the full moon was our earth. The scene was 
not unlike that shown to Satan when Uriel, 

" one of the seven 
Who in God's presence, nearest to the throne, 
Stand ready at command," 

pointing earthwards from his station amid the splen- 
dour of the sun, said to the archfiend, 

" Look downward on that globe whose hither side 
With light from hence, though but reflected, shines : 
That place is earth, the seat of man ; that light 
His day, which else, as th' other hemisphere, 
Night would invade." 

In all other respects the scene presented to the 
spectator on the moon was similar; but as seen from 
the lunar Apennines the glorious orb of earth shone 
high in the heavens ; and the sun, source of the light 
then bathing her oceans and continents, lay far down 
below the level of the lunar horizon. 

And now, as hour passed after hour, a series of 
changes took place in the scene, which were unlike 
any that are known to our astronomers on earth. 
The stars passed, indeed, athwart the heavens on a 
course not differing from that followed by the stars 
which illumine our skies, but so slowly that in an 
hour of lunar time they shifted no more than our stars 
do in about two minutes. And marvellous to see, 
the great orb of earth did not partake in this motion. 
Hour by hour passed away, the stars slowly moved 
on their course westwards, but they left the earth still 



suspended as a vast orb of light high above the 
southern horizon. She changed, indeed, in aspect. 
The two Americas passed away towards the right, 
and the broad Pacific was presented to view. Then 
Asia and Australia appeared on the left, and as they 
passed onwards the East Indies came centrally upon 
the disc. Then the whole breadth of Asia could be 
recognized, but partly lost in the misty light of the 
northern half, while the blue of the Indian Ocean was 
conspicuous in the south. And as the hours passed 
on, Europe and Africa came into view, and our own 
England, foreshortened and barely visible, near the 
snow-covered northern region of the disc. 

But although such changes as these took place in 
the aspect of the earth, her globe remained almost 
unchanged in position. It was indeed traversing the 
ecliptical zone, along which the sun and moon and 
planets pursue their course; and this star-zone was 
itself being carried slowly round the lunar sky : but 
these motions were so adjusted that the earth herself 
appeared at rest. The zone of the ecliptic was carried 
round from east to west behind the almost unmoving 
globe of the earth. When South America was in view, 
she had been close to the eastern border of Aries; 
and now Aries had passed away westwards, and Taurus 
was behind the earth. And yet it could not be said 
that the earth by advancing along the ecliptic was 
hiding the stars of the zodiacal constellations ; rather 
it appeared as though these stars were hiding them- 
selves in turn behind the earth. 


But the stars were not hidden as they are when 
the moon passes over them. The terrestrial astro- 
nomer in such a case observes that a star vanishes 
instantly^ and reappears with equal suddenness when 
the due time has arrived. But the passage of the 
multitudinous stars of the lunar sky behind the earth 
was accomphshed in a different manner. The border 
of the earth^s disc was seen to be full of a light far 
more resplendent than that of the disc itself. As the 
stars on their passage to the region behind the earth 
approached this border, their light was seen to be 
merged in the ring of splendour. This ring was, in 
fact, produced by the mingled lustre of all the stars 
which were behind the earth's disc; and speaking 
correctly these stars did not vanish at all. The 
earth's atmosphere, like a gigantic lens, brought all 
these stars into view, and became filled with their 
diffused light, just as the object-glass of a telescope is 
seen to be filled with a star's light when we remove 
the eyeglass. 

Here then was another feature in which the earth, 
seen as a celestial body from the moon, differed wholly 
from any celestial orb visible to terrestrial astronomers. 
Her orb, beautiful from its size and splendour, beau- 
tiful also in its variegated colours, was girt around 
with a ring of star-light, a ring infinitely fine as seen 
from the moon by vision such as ours, yet conspicuous 
because of the quality of the light which produced it. 

It may seem surprising that though the orb of 
earth was shining so splendidly above the lunar 
Y 2 


horizon^ stars could be seen which the far fainter 
lustre of the full moon obliterates from our skies ; and 
not these alone, but countless thousands of other stars, 
which only the telescopist can see from a terrestrial 
station. But the observer on the moon has no sky, 
properly so called. Above and around him is the 
vault of heaven, while the atmosphere which forms 
our sky, not only in the splendour of day, but in the 
darkest night, when the stars seem to shine as on a 
background of intense blackness, is wanting on the 
moon. The blackness of our darkest skies is as the 
light of day by comparison with the darkness of 
space on which the stars of the lunar heavens are seen 
projected. The glorious orb of the earth was there at 
the time we speak of, and her light would have lit up 
an atmosphere like our own, so that the whole sky 
would have been aglow ; but on the moon there w^as 
no atmosphere to illuminate, so that above and around 
the observer there was no sky. 

Yet the lunar lands were lit up with the splen- 
dour of earth-light. The mountainous region around 
shone far more brightly than a similar terrestrial 
scene under the full moon, and the glory of the 
earth-lit portions was rendered so much the more 
remarkable by the amazing blackness of the parts 
which were in shadow. But the lustre of the stars 
was not dimmed. There was no veil of light to hide 
the stars, as when the full moon pours her rays upon 
the terrestrial air. Homer^s famous description of a 
moonht night corresponds far better with the lunar 


scene than with, night on the earth. For whereas 
on earth the glory of the moon hides the heaven of 
stars from our view^ on the moon, in the far greater 
splendour of the full earth, 

" the stars about the Earth 

Look beautiful 

And every height comes out, and jutting peak 
And valley, and the immeasurable heavens 
Break open to their highest, and all the stars 

The long hours passed, measured by the stately 
motion of the stars behind the scarce moving earth, 
and by the changing aspect of her globe, as continents 
and oceans were carried from left to right across her 
face by her rotation. And gradually her orb lost its 
roundness. The ring of brilliant star-light which 
encircled her disc remained perfectly round indeed, 
but within this ring on the right a dark sickle began 
to be seen, and, slowly spreading, invaded the disc on 
that side. The earth was no longer full, but had 
assumed a gibbous phase, like the moon a day or two 
after full. Yet her aspect was wholly unlike that of 
the gibbous moon. The ring of light surrounding her 
true orb would of itself have made her appear unlike 
the moon ; but besides this peculiarity, tnere was a 
marked contrast in the appearance of the darkened 
portion. Instead of that sharply- defined edge pre- 
sented by the gibbous moon, there was in the case of 
the earth a softening off of the light by gradations 
so gentle that no eye could perceive where the en- 


lightened hemisphere terminated and the dark hemi- 
sphere began. As night on our earth comes on with 
stealthy pace^ the shades of evening closing in so 
gradually that we can hardly say when day ends and 
night begins, so from the station of the lunar observer 
the shading on the earth's darkened side which showed 
where night was coming on presented no recognizable 
outlines. One familiar with the earth and with the 
ways of her inhabitants could not but picture to 
himself how, as country after country was carried by 
the earth's rotation into that darkened region, the 
labours of men were being drawn towards a close for 
the day. 

When about a week had passed, the earth had 
become a half-earth. The shape of the darkened half 
of the disc could still be recognized by the ring of 
star-light, which always surrounds her as she is seen 
from the moon, and remains nearly always bright 
and conspicuous, though sometimes, when many and 
bright stars are behind the earth, the ring is brighter 
than at others. This in fact had been the case when 
the Milky Way, where it crosses Gemini, had been 
carried behind the earth, which now, however, had 
passed beyond that region, and was entering the 
constellation of the Lion. 

The aspect of the earth had in the meanwhile 
altered in another respect. The southern polar regions 
had been turned more fully than before towards the 
moon, more fully than towards the sun. 

We may pass, however, from the further con- 


sideration of changes in the earth's aspect, to describe 
a far more interesting series of phenomena which had 
already commenced to be discernible in the eastern 
part of the heavens. 

At all times the zodiacal light is visible in the lunar 
heavens, forming a zone completely round the zodiac, 
and perfectly distinct in appearance from the Milky 
Way. It is far more brilliant, even when faintest, 
than the zodiacal light we recognize through our air, 
at once dense enough to conceal, and sufficiently illu- 
minated, whether by twilight, moonlight, or starlight, 
to spread a veil over the delicate light of the zodiacal. 
But near the sun's place the zodiacal has an aspect 
utterly unlike that of even the brightest portions 
seen by us. Its complicated structure becomes dis- 
cernible, and its colour indicates its community of 
nature with the outer parts of the solar corona. At 
the epoch we are considering, the corona itself was 
rising in the east, and its outer streamers could be 
seen extending along the ecliptical zone far into the 
bright core of the zodiacal. 

Infinitely more wonderful, however, and transcend- 
ing in sublimity all that the heavens display to the 
contemplation of the inhabitants of earth, was the 
scene presented when the sun himself had risen. I 
shall venture here to borrow some passages from an 
essay entitled "A Yoyage to the Sun,'' in which a friend 
of mine has described the aspect of the sun as seen 
from a station outside that atmosphere of ours which 
veils the chief glories of the luminary of day. " The 


sun^s orb was more brilliantly white than when seen 
through the air, but close scrutiny revealed a diminution 
of brilliancy towards the edge of the disc, which, when 
fully recognized, presented him at once as the globe 
he really is. On this globe could be distinguished 
the spots and the bright streaks called faculae. This 
globe was surrounded with the most amazingly com- 
plex halo of glory. Close around the bright whiteness 
of the disc, and shining far more beautiful by contrast 
with that whiteness than as seen against the black 
disc of the moon in total eclipses stood the coloured 
region called the chromatosphere, not red, as it 
appears during eclipses, but gleaming with a mixed 
lustre of pink and green, through which, from time to 
time, passed the most startlingly brilliant coruscations 
of orange and golden yellow light. Above this delicate 
circle of colour towered tall prominences and multitudes 
of smaller ones. These, like the chromatosphere, were 
not red, but beautifully variegated. In parts of the 
prominences colours appeared which were not seen in 
the chromatosphere,' and in particular, certain blue 
and purple points of light which were charmingly 
contrasted with the orange aud yellow flashes con- 
tinually passing along the whole length of even the 
loftiest of these amazing objects. The prominences 
round different parts of the sun^s orb presented very 
different appearances ; for those near the sun's equa- 
torial zone and opposite his polar regions differed very 
little in their colour and degree of light from the 
chromatosphere. They also presented shapes re- 


sembling rather those of clouds moving in a perturbed 
atmosphere, than those which would result from the 
tremendous processes of disturbance which astro- 
nomers have lately shown to be in progress in the 
sun. But opposite the spot-zones the prominences 
presented a totally different appearance. They re- 
sembled jets of molten matter, intensely bright, 
and seemingly moving with immense velocity. They 
formed and vanished with amazing rapidity, as when 
in terrestrial conflagrations a flame leaps suddenly to 
a great height and presently disappears.^^./^ Around 
the sun a brightly luminous envelope extended to about 
twice the height of the loftiest prominences, while above 
even the faintest signs of an atmosphere, as well as 
through and amidst both the inner bright envelope 
and the fainter light surrounding it, there were the 
most complex sprays and streams and filaments of 
whitish light, here appearing as streamers, elsewhere 
as a network of bright streaks, and yet elsewhere 
clustered into aggregations which can be compared to 
nothing so fitly, though the comparison may seem 
commonplace, as to hanks of glittering thread. All 
these streaks and sprays of light appeared perfectly 
white, and only differed among themselves in that 
whereas some appeared like fine streaks of a uniform 
silvery lustre, others seemed to shine with a curdled 
light. The faint light outside the glowing atmosphere 
surrounding the prominences was also whitish ; but 
the glowing atmosphere itself shone with a light 
resembling that of the chromato sphere, only not so 


brilliant. The pink and green lustre, continually 
shifting, so that a region which appeared pink at one 
time would shine a short time aftr with a greenish, 
light, might aptly be compared in appearance to 
mother-of-pearl. The real extension of the white 
streaks and streamers was not distinguishable, for 
they became less and less distinct at a greater and 
greater distance from the sun, and finally became 

Much more might be said on this inviting subject, 
only that the requirements of space forbid, obliging 
me to remember that the moon and not the sun is the 
subject of this treatise. The reader, therefore, must 
picture to himself the advance of the sun with his 
splendid and complicated surroundings towards the 
earth, suspended almost unchangingly in the heavens, 
but changing gradually into crescent form as the sun 
drew slowly near. He must imagine also, how, in the 
meantime, the star-sphere was slowly moving west- 
wards, the constellations of the ecliptic in orderly 
succession passing behind the earth at a rate slightly 
exceeding that of the sun's approach, so that he, like 
the earth, only more slowly, was moving eastwards so 
far as the star-sphere was concerned, even while the 
moon's slow diurnal rotation was carrying him west- 
wards towards the earth. 

At the station we are considering, the lunar eclipse 
which took place on November 15, 1872, was only 
partial. Here, therefore, though the sun actually 
passed in part behind the earth, a portion of his orb 


remained unconcealed. But owing to the refractive 
power of the earth's atmosphere the rest of his disc 
was also brought into view, amazingly distorted, and 
forming a widely-extended crescent of red light true 
sun-light around a large arc of the earth's edge, the 
visible portion of the solar disc being at the middle of 
this crescent. 

To an observer near the north pole of the moon, the 
eclipse was total, at least in our terrestrial mode of 
considering lunar eclipses : the true shadow of the 
earth fell on that portion of the moon. From a station 
so placed then, no part of the sun's disc could be seen 
by the lunarian ; nevertheless a crescent of sun-light 
was visible in this case also, the crescent extending 
farther round the earth's disc than in the former case, 
and in fact round considerably more than a semicircle, 
the brightest part of the crescent being opposite the 
part of the earth's disc behind which the sun's disc 
was in reality placed. 

I must, however, leave the reader to conceive the 
slow processes of change by which, as the sun ad- 
vanced to the position here indicated, his disc became 
gradually modified into this crescent of true sun- 
light, this distorted image of the wJiole sun, thus seen 
through the spherical shell of the earth's atmo- 
sphere, and how, passing onwards towards the west, 
he gradually reappeared. Space will not permit me 
to dwell as I should wish on the multitude of interest- 
ing relations presented as the solar surroundings passed 
in their turn behind the earth, either before the sun as 


lie approached the earth, or after the sun as he moved 
on westwards. Enough has been said to indicate to 
the thoughtful reader the general nature of the pheno- 
mena presented during the whole course of the sun's 
passage from the eastern to the western horizon, as 
well as those which followed after he had set, until the 
lunar month was complete, and the earth again seen, 
on November 30, 1872, with fully illuminated orb 
upon the lunar sky. 





If the study of our earth's crust or the science of 
geology is capable of throwing some degree of light 
on the past condition of other members of the solar 
system, the study of those other orbs seems capable 
of at least suggesting useful ideas concerning the past 
condition of our earth. There are members of the 
solar system respecting which it may reasonably be 
inferred that they are in an earlier stage of their 
existence than the earth. Jupiter and Saturn j for 
instance, would seem so far as observation has ex- 
tended to be still in a condition of intense heat, and 
still the seat of forces such as were once probably at 
work within our earth. "We see these planets en- 
wrapped, to all appearance, within a double or triple 
coating of clouds, and we are compelled to infer, from 
the behaviour of these clouds, that they are generated 
by forces belonging to the orb which they envelope ; 
we have, also, every reason which the nature of the 

* This chapter is reprinted, with slight alterations, from the 
Quarterly Journal of Science, "by the kind permission of the editor 
of that serial. 

334 CONDITION or the moon^s surface. 

case can afford, to suppose that our own earth was 
once similarly cloud- enveloped. We can scarcely 
imagine that in the long-past ages, when the igneous 
rocks were in the primary stages of their existence, 
the air was not loaded heavily with clouds. We may, 
then, regard Jupiter and Saturn as to some degree in- 
dicating the state of our own earth at a long-past 
epoch of her existence. On the other hand, it has 
been held, and not without some degree of evidence 
in favour of the theory, that in our moon we have a 
picture of our earth as she will be at some far-distant 
future date, when her period of rotation has been 
forced into accordance with the period of the moon's 
revolution round the earth, when the internal heat of 
the earth's globe has been radiated almost wholly 
away into space, and when her oceans and atmosphere 
have disappeared through the action of the same cir- 
cumstances (whatever they may be) which have caused 
the moon to be air-less and ocean-less. But whether 
we take this view of our earth's future, or whether we 
consider that her state has been from the beginning 
very different from that of the moon, it nevertheless 
remains probable that we see in our moon a globe 
which has passed through a much greater proportion 
of its history (so to speak) than our earth ; and accord- 
ingly the study of the moon's condition seems capable 
of giving some degree of information as to the future 
(possibly also as to the past) of our earth. 

I wish here to consider the moon's condition from 
a somewhat different point of view than has commonly 


been adopted. It appears to me that tlie study of the 
moon^s surface with the telescope, and the considera- 
tion of the various phenomena which give evidence on 
the question whether air or water exist anywhere upon 
or within her, have not as yet led to any satisfactory 
inferences as to her past history. We see the traces 
of tremendous sublunarian disturbances (using the 
word ^sublunarian/ here and elsewhere, to corre- 
spond to the word ' subterranean ' used with reference 
to the earth), and we find some features of resemblance 
between the effects of such disturbances and those 
produced by the subterranean forces of our earth ; but 
we find also as marked signs of distinction between 
the features of the lunar and terrestrial crusts. Again, 
comparing the evidences of a lunar atmosphere with 
those which we should expect if an atmosphere like 
our own surrounded the moon, we are able to decide, 
with some degree of confidence, that the moon has 
either no atmosphere or one of very limited extent. 
But there our knowledge comes to an end; nor 
does it seem likely that, by any contrivances man 
can devise, the further questions which suggest them- 
selves respecting the moon's condition can be answered 
by means of observation. 

Yet there are certain considerations respecting the 
moon's past history which seem to me likely, if duly 
weighed, to throw some light on the difficult problems 
presented by the moon. 

In the first place, it is to be noted that the pecuhar 
relation between the moon's rotation and revolution 


possesses a meaning which has not hitherto, so far as I 
know, been attended to. We know that noiv there is an 
absolutely perfect agreement between the moon's rota- 
tion and revolution, in this respect, that her mean 
period of rotation on her axis is exactly equal to her 
mean period of revolution. (Here either sidereal rota- 
tion and revolution or sy nodical rotation and revolution 
may be understood, so long as both revolution and 
rotation are understood to be of the same kind.) I 
say " mean period of rotation,'^ for although as a matter 
of fact it is only the revolution which is subject to any 
considerable variation, the rotation also is not perfectly 
uniform. "We know, furthermore, that if there had 
been, long ago, a near agreement between the mean 
rotation and revolution, the present exact agreement 
would have resulted, through the effects of the mutual 
attractions of the earth and moon. But so far as I 
know, astronomers have not yet carefully considered 
the question whether that close agreement existed 
from the beginning, or was the result of other forms 
of action than are at present at work. If it existed 
from the beginning, that is from the moon's first 
existence as a body independent of the earth, it is a 
matter requiring to be explained, as it implies a pecu- 
liar relation between the moon and earth before the 
present state of things existed. If, on the contrary, 
it has been brought about by the amount of action 
which is now gradually reducing the earth's rotation- 
period, we have first of all to consider that an enor- 
mous period of time has been required to bring the 


moon to her present condition in this respect^ and 
moreover,, that either an ocean existed on her surface 
or that her crust was once in so plastic a condition as 
to be traversed by a tidal wave resembling, in some 
respects, the tidal wave in our own ocean. This, at 
any rate, is what we must believe if we suppose, first, 
that the main cause of the lengthening of the terres- 
trial day is the action of the tidal wave as a sort of 
brake on the earth^s rotating globe, and secondly, 
that a similar cause produced the lengthening of the 
moon's day to its present enormous duration. It may 
be, as we shall presently see, that other causes have 
to be taken into account in the moon's case. 

Now we are thus, either way, brought to a consider- 
ation of that distant epoch when according to the 
nebular theory, or any admissible modification thereof 
the moon was as yet non-existent as an orb distinct 
from the earth. We must suppose, on one theory, 
that the moon was at that time enveloped in the 
nebulous rotating spheroid out of which the earth was 
to be formed, she herself (the moon) being a nebulous 
sub-spheroid within the other, and so far coerced by 
the motion of the other that her longer axis partook 
in its motion of rotation. Unquestionably in that 
case, as the terrestrial spheroid contracted and left the 
other as a separate body, this other, or lunar spheroid, 
would exhibit the kind of rotation which the moon 
actually possesses. On the other theory, we should 
be led to suppose that primarily the lunar spheroid 
rotated independently of its revolution ; but that the 



earth^s attraction acting on the outer shells, after 
they had become first fluid and then (probably) 
viscous, produced waves travelling in the same direc- 
tion as the rotation, but with a continual brake-action, 
tending slowly to reduce the rotation until it had its 
present value, when dynamical equilibrium would be 

But, as I have said, in either case we must trace 
back the moon^s history to an epoch when she was in 
a state of intense heat. And it seems to me that we 
are thus led to notice that the development of the 
present state of things in the moon must have taken 
place during an era in the history of the solar system 
differing essentially from that which prevailed during 
the later and better-known geological eras of our own 
earth. Our moon was shaded, so to speak, when the 
solar system itself was young, when the sun may have 
given out a much greater degree of heat than at 
present, when Saturn and Jupiter were brilliant suns, 
when even our earth and her fellow minor planets 
within the zone of asteroids were probably in a sun- 
like condition. Putting aside all hypothesis, it never- 
theless remains clear that, to understand the moon's 
present condition, we must form some estimate of the 
probable condition of the solar system in distant eras 
of its existence ; for it was in such eras, and not in an 
era like the present, that she was modelled to her 
present figure. 

It appears to me that we are thus, to some extent, 
freed from a consideration which has proved a diffi- 


ciilty to many wlio have theorized respecting the 
moon. It has been said that the evidence of volcanic 
action implies the existence, at least when that action 
was in progress, of an atmosphere capable of sup- 
porting combustion, in other words, an atmosphere 
containing oxygen, for other forms of combustion than 
those in which oxygen plays a part may here be 
dismissed from consideration. But the fiery heat of 
the moon's substance may have been maintained (in 
the distant eras to which we are now referring 
the formation of her crust) without combustion. 
Taking the nebular hypothesis as it is commonly 
presented, the moon's globe may have remained 
amid the intensely hot nebulous spheroid (which was 
one day to contract, and so form the globe of the 
earth) until the nebula left it to cool thenceforth 
rapidly to its present state. Whatever objections 
suggest themselves to such a view are precisely the 
objections which oppose themselves to the simple 
nebluar hypothesis, and may be disposed of by those 
who accept that hypothesis. But better, to my view, 
it may be reasoned, that the processes of contraction 
and of the gathering in of matter from without, which 
maintained the heat of the nebulous masses, operated 
to produce all the processes of disturbance which 
brought the moon to her present condition, and that 
thus there was not necessarily any combustion what- 
ever. Indeed, in any case, combustion can only have 
commenced when the heat had been so far reduced 
that any oxygen existing in the lunar spheroid would 
z 2 


enter into chemical combination with various com- 
ponents of the moon's glowing substance. If there 
were no oxygen (an unlikely supposition^ however), 
the moon's heat would nevertheless have been main- 
tained so long as meteoric impact on the one hand_, 
and contraction of the moon's substance on the other, 
continued to supply the requisite mechanical sources 
of heat-generation. In this case there would not 
necessarily have been any gaseous or vaporous 
matter, other than the matter retained in the gaseous 
condition by intensity of heat, and becoming first 
liquid and afterwards solid so soon as the heat was 
sufficiently reduced. 

It must here be considered how far we have reason 
to believe that the heat of the various members of the 
solar system including the moon and other secondary 
bodies was originally produced, and thereafter main- 
tained, by collisions ; because it is clear that, as 
regards the surface contour of these bodies, much, 
would depend on this circumstance. There would be 
a considerable difference between the condition of a 
body which was maintained at a high temperature for 
a long period, and eventually cooled, but slowly, 
nnder a continual downfall of matter, and that of a 
body whose heat was maintained by a process of 
gradual contraction. It is true that in the case of a 
globe like the earth, whose surface was eventually 
modelled and re-modelled by processes of a totally 
different kind, by deposition and denudation, by wind 
and rain, river-action and the beating of seas, the 


signs of tlie original processes of cooling would to a 
great extent disappear ; but if, as we are supposing 
in the case of the moon, there was neither water 
nor air (at least in sufficient quantity to produce any 
effect corresponding to those produced by air and 
water on the earth) , the principal features of the 
surface would depend largely on the conditions under 
which the process of cooling began and proceeded. 

Now here I must recall to the attention of the 
reader the reasoning which I have made use of in my 
" Other Worlds than Ours_,^^ to show that_, in all pro- 
babihty, our solar system owed its origin rather to 
the gathering of matter together from outer space 
than to the contraction of a rotating nebulous mass. 
It is there shown, and I think that the consideration 
is one which should have weight in such an inquiry, 
that there is nothing in the nebular hypothesis of 
Laplace to account in any degree for the peculiarities 
of detail presented by the solar system. That theory 
explains the revolution of the members of the solar 
system in the same direction, their rotation in the same 
direction, the approach to circularity of the orbits, 
and their near coincidence with the mean plane of the 
system ; but it leaves altogether unexplained the 
different dimensions of the primary members of the 
solar system, the apparent absence of law and order 
in their axial tilt and the inclination of the orbits of 
their satellite families. In particular, the remarkable 
difference which exists between the outer family of 
planets, the giant orbs, Jupiter, Saturn, Uranus, and 


Neptune^ and the inner family of small planets, 
Mars, the Earth, Yenus, and Mercury, is left wholly 
unexplained. Nor can one recognize in the nebular 
hypothesis any reason whatever for* the comparative 
exuberance of orb-forming activity in the outer family, 
and particularly in the two planets lying next to the 
zone of asteroids, and the poverty of material which 
is exhibited within the minor family of planets. All 
these circumstances appear to be explained satis- 
factorily when we regard the solar system as formed 
by the gathering in from outer space of materials 
once widely scattered. We can see that in the 
neighbourhood of the great primary centre there 
would be indeed a great abundance of gathered and 
gathering matter, but that, owing to the enormous 
velocities in that neighbourhood, subordinate centres 
of attraction would there form slowly, and acquire but 
moderate dimensions. Outside a certain distance there 
would be less matter, but a far greater freedom of 
aggregation; there we should find the giant secondary 
centres, and we should expect the chief of these to 
lie inwards, as Jupiter and Saturn, while beyond 
would be orbs vast indeed, but far inferior to these 
planets. And we can readily see that the border 
region between the family of minor planets and the 
family of major planets would be one where the 
formation of a planet would be rendered unlikely; 
here, therefore, we should look for the existence of a 
zone of small bodies like the asteroids. I touch on 
these points to show the kind of evidence (elsewhere 


given at length) on wliicli I have based my opinion 
that the solar system had its birth_, and long main- 
tained its fires_, under the impact and collisions of 
bodies gathered in from outer space. 

According to this view, the moon, formed at a 
comparatively distant epoch in the history of the 
solar system, would have not merely had its heat 
originally generated for the most part by meteoric 
impact, but while still plastic would have been ex- 
posed to meteoric downfalls, compared with which all 
that we know, in the present day, of meteor- showers, 
aerolitic masses, and so on, must be regarded as 
altogether insignificant. It would be to such down- 
fall mainly that the maintenance of the moon^s heat 
would at that time be due, though, as we shall pre- 
sently see, processes of contraction must have not 
only supplemented this source of heat-supply, but 
must have continued to maintain the moon^s heat 
long after the meteoric source of heat had become 
comparatively ineffective. 

Now, I would notice in passing that here we may 
find an explanation of the agreement between the 
moon's rotation-period and her period of revolution. 
It is clear that under the continuous downfall of 
meteoric matter in that distant era, the moon must 
have been in a process of actual growth. She is 
indeed growing noiu from the same cause ; and so is 
the earth : but such growth must be regarded as 
infinitesimally small. In the earlier periods of the 
moon's history, on the contrary, the moon's growth 


must have progressed at a comparatively rapid rate. 
Now this influx of matter must have resulted in a 
gradual reduction of the moon's rate of rotation, if (as 
we must suppose) the moon gathered matter merely 
by chance collisions. In the case of a globe gather- 
ing in matter by its own attractive power, as the sun 
does for instance, the arriving matter may (owing to 
the manner in which the process is effected) serve to 
maintain and even to increase the rate of rotation; 
but in the case of a subordinate body like the moon 
we must suppose that all effects acting on the rotation 
would be about equally balanced, and that the sole 
really effective result would be the increase of the 
moon's bulk, and the consequent diminution of her 
rotation-rate. Now, if this process continued until 
the rotation-rate had nearly reached its present value, 
the earth's attraction would suffice not merely to 
bring the rate of rotation precisely to its present 
value, but to prevent its changing (by the continuance 
of the process) to a smaller value. It may be added 
that the increase in the moon's rate of revolution, as 
she herself and the earth both grew under meteoric 
downfall towards their ' present dimensions, would 
operate in a similar way, it would tend to bring 
the moon's rate of revolution and her rate of 
rotation towards that agreement which at present 

If we attempt to picture the condition of the moon 
in that era of her history when first the process of 
downfall became so far reduced in activity as to permit 


of her cooling down, \ye shall be tempted, I believe, 
to consider that some of the more remarkable features 
of her globe had their origin in that period. It may 
seem, indeed, at a first view, too wild and fanciful an 
idea to suggest that the multitudinous craters on the 
moon, and especially the smaller craters revealed in 
countless numbers when telescopes of high power are 
employed, have been caused by the plash of meteoric 
rain, and I should certainly not care to maintain 
that as the true theory of their origin; yet it must 
be remembered that no plausible theory has yet been 
urged respecting this remarkable feature of the 
moon^s surface. It is impossible to recognize a real 
resemblance between any terrestrial feature and the 
crateriferous surface of the moon. As blowholes, so 
many openings cannot at any time have been neces- 
sary, whatever opinion we may form as to the con- 
dition of the moon^s interior and its reaction upon the 
crust. Moreover, it should be remembered that our 
leading seismologists regard water as absolutely es- 
sential to the production of volcanic disturbance (the 
only form of disturbance which on our earth leads to 
the formation of cup-shaped openings) . If we consider 
the explanation advanced by Hooke, that these nume- 
rous craters were produced in the same way that 
small cup-shaped depressions are formed when thick 
calcareous solutions are boiled and left to cool, we see 
that it is inadequate to account for lunar craters, the 
least of which (those to which Mr. Birt has given the 
name of craterlets) are at least half a mile in diameter. 


The rings obtained by Hooke were formed by the 
breaking of surface bubbles or Misters,* and it is 
impossible for such bubbles to be formed on the scale 
of the lunar craters. Now so far as the smaller craters 
are concerned, there is nothing incredible in the sup- 
position that they were due to meteoric rain falling 
when the moon was in a plastic condition. Indeed, it 
is somewhat remarkable how strikingly certain parts 
of the moon resemble a surface which has been rained 
upon while sufficiently plastic to receive the im- 
pressions, but not too soft to retain them. Nor is it 
any valid objection to this supposition, that the rings 
left by meteoric downfall would only, be circular when 
the falling matter chanced to strike the moon^s sur- 
face squarely; for it is far more probable that even 
when the surface was struck very obliquely and the 
opening first formed by the meteoric mass or cloud 
of bodies was therefore markedly elliptic, the plastic 
surface would close in round the place of impact until 
the impression actually formed had assumed a nearly 
circular shape. 

Before passing from this part of my subject, I would 
invite attention to the aspect of the moon as presented 
in the photographs illustrating this work. It will be 

* " Presently ceasing to boil," he says of alabaster, " the whole 
surface will appear covered all over with small pits, exactly shaped 
like those of the moon." " The earthy part of the moon has been 
undermined," he proceeds, " or heaved up by eruptions of vapour, 
and thrown into the same kind of figured holes as the powder of 


seen tliat tlie multitudinous craters near the top of these 
pictures (the southern part of the moon) are strongly 
suggestive of the kind of process I have referred to, 
and that, in fact^ if one judged solely by appearances, 
one would be disposed to adopt somewhat confidently 
the theory that the moon had had her present surface 
contour chiefly formed by meteoric downfalls during 
the period of her existence when she was plastic to 
impressions from without. I am, however, sensible 
that the great craters under close telescopic scrutiny 
by no means correspond in appearance to what we 
should expect if they were formed by the downfall of 
great masses from without. The regular, and we may 
almost say battlemented, aspect of some of these 
craters, the level floor, and the central peaks so 
commonly recognized, seem altogether diff'erent from 
what we should expect if a great mass fell from outer 
space upon the moon^s surface. It is indeed just 
possible that under the tremendous heat generated 
by the downfall, a vast circular region of the moon^s 
surface would be rendered liquid, and that in rapidly 
solidifying while still traversed by the ring-waves 
resulting from the downfall, something like the present 
condition would result. Or we might suppose that 
the region liquefied through the efiects of the shock 
was very much larger than the meteoric mass ; and 
that while a wave of disturbance travelled outwards 
from the place of impact, to be solidified (owing to 
rapid radiation of heat) even as it travelled, a portion 
of the liquid interior of the moon forced its way through 


the opening formed by the falling mass. But such 
ideas as these require to be supported by much 
stronger evidence than we possess before they can be 
regarded as acceptable. I would remark, however, 
that nothing hitherto advanced has explained at all 
satifactorily the structure of the great crateriform 
mountain-ranges on the moon. The theory that there 
were once great lakes seems open to difficulties at 
least as grave as the one I have just considered, and 
to this further objection, that it affords no explanation 
of the circular shape of these lunar regions. On the 
other hand. Sir John HerschePs account of the ap- 
pearance of these craters is not supported by any 
reasoning based on our knowledge of the actual 
circumstances under which volcanic action proceeds 
in the case of our own earth. " The generality of the 
lunar mountains," he says, ""present a striking 
uniformity and singularity of aspect. They are won- 
derfully numerous, occupying by far the larger portion 
of the surface, and almost universally of an exact 
circular or cup-shaped form, foreshortened, however, 
into ellipses towards the limb ; but the larger have for 
the most part flat bottoms within, from which rises 
centrally a small, steep, conical hill. They offer, in 
short, in its highest perfection, the true volcanic 
character, as it may be seen in the crater of Vesuvius ; 
and in some of the principal ones, decisive marks of 
volcanic stratification, arising from successive deposits 
of ejected matter, may be clearly traced with powerful 
telescopes. What is, moreover, extremely singular in 


the geology of the moon is^ that although nothing- 
having the character of seas can be traced (for the 
dusty spots which are commonly called seas, when 
closely examined, present appearances incompatible 
with the supposition of deep water), yet there are 
large regions perfectly level, and apparently of a de- 
cided alluvial character." 

It is obvious that in this description we have, 
besides those features of volcanic action which might, 
perhaps, be expected on the moon, a reference to 
features essentially terrestrial. Alluvial deposits can 
have no existence, for example, save where there are 
rivers and seas, as well as an atmosphere within which 
clouds may form, whence rain may be poured upon the 
surface of wide land regions. It is not going too far 
to say that we have the clearest evidence to show that 
in the moon none of these conditions are fulfilled. 
Whether in former ages lunar oceans and seas and a 
lunar atmosphere have existed, may be a doubtful 
point ; but it is certain that all the evidence we have 
is negative, save only those extremely doubtful signs 
of glacier action recognized by Professor Frankland. 
I venture to quote from Guillemin's " Heavens " a 
statement of Frankland' s views, in order that the 
reader may see on how slender a foundation hypo- 
theses far more startling than the theory I have 
suggested have been based by a careful reasoner and 
able physicist. " Professor Frankland believes," says 
the account, '' and his belief rests on a special study 
of the lunar surface, that our satellite has, like its 


primary^ also passed through a glacial epoch^ and that 
several^ at leasts of the valley s, rills, and streaks of the 
lunar surface are not improbably due to former glacial 
action. Notwithstanding the excellent definition of 
modern telescopes, it could not be expected that other 
than the most gigantic of the characteristic details 
of an ancient glacier-bed would be rendered visible. 
Whatj then_, may we expect to see ? Under favour- 
able circumstances, the terminal moraine of a glacier 
attains enormous dimensions ; and consequently, of all 
the marks of a glacier valley, this would be the one 
most likely to be first perceired. Two such terminal 
moraines, one of them a double one, have appeared to 
observers to be traceable upon the moon's surface. 
The first is situated near the termination of the re- 
markable .streak which commences near the base of 
Tycho, and passing under the south-eastern wall of 
Bullialdus, into the ring of which it appears to cut, is 
gradually lost after passing Lubiniezky. Exactly 
opposite this last, and extending nearly across the 
streak in question, are two ridges forming the arcs 
of circles whose centres are not coincident^ and whose 
external curvature is towards the north. Beyond the 
second ridge a talus slopes gradually down northwards 
to the general level of the lunar surface, the whole 
presenting an appearance reminding the observer of 
the concentric moraines of the Rhone glacier. These 
ridges are visible for the whole period during which 
that portion of the moon's surface is illuminated ; but 
it is only about the third day after the first quarter. 


and at the corresponding phase of the waning moon, 
when the sun^s rays, falling nearly horizontally, throw 
the details of this part of the surface into strong relief, 
and these appearances suggest this explanation of 
them. The other ridge answering to a terminal 
moraine, occurs at the northern extremity of that 
magnificent valley which runs past the eastern edge of 

Here are two lunar features of extreme delicacy, 
and certainly not incapable of being otherwise ex- 
plained, referred by Frankland to glacier action. It 
ifeed hardly be said that glacial action implies the 
existence of water and an atmosphere on the moon ; 
and not only so, but there must have been extensive 
oceans, and an atmosphere nearly equal in density to 
tbat of our own earth, if the appearances commented 
upon by Frankland were due to glacial action. It is 
admitted by Frankland, of course, that there is now 
no evidence whatever of the presence of water, " but, 
on the contrary, all selenographical observations tend 
to prove its absence. Nevertheless," proceeds the 
account from which I have already quoted, " the idea 
of former aqueous agency in the moon has received 
almost universal acceptation^' (the italics are mine). 
^' It was entertained by Gruithuisen and others. But, 
if water at one time existed on the surface of the moon, 
whither has it disappeared ? If we assume, in ac- 
cordance with the nebular hypothesis, that the portions 
of matter composing respectively the earth and the 
moon once possessed an equally elevated temperature^ 


it almost necessarity follows that the moon^ owing to 
the comparative smallness of her mass, would cool 
more rapidly than the earth ; for whilst the volume of 
the moon is only about 1 -49th (and its mass, it might 
be added^ only about 1-81 st part), its surface is nearly 
1-1 3th that of the earth. This cooling of the mass of 
the moon must, in accordance with all analogy, have 
been attended with contraction, which can scarcely be 
conceived as occurring without the development of a 
cavernous structure in the interior. Much of this 
cavernous structure would doubtless communicate, by 
means of fissures, with the surface, and thus ther^ 
would be provided an internal receptacle for the ocean, 
from the depths of which even the burning sun of the 
long lunar day would be totally unable to dislodge 
more than traces of its vapour. Assuming the solid 
mass of the moon to contract on cooling at the same 
rate as granite, its refrigeration though only 180 F. 
would create cellular space equal to nearly fourteen 
and a half millions of cubic miles, which would be 
more than sufficient to engulf the whole of the lunar 
oceans, supposing them to bear the same proportion 
to the mass of the moon as our own oceans bear to 
that of the earth." 

The great objection to this view of the moon^s past 
history consists in the difficulty of accounting for the 
lunar atmosphere. It must be remembered that 
ov/ing to the smallness of the moon's mass, an atmo- 
sphere composed in the same way as ours would have- 
a much greater depth compared with its density at 


the mean level of the nioon\s surface than our atmo- 
sphere possesses compared with its pressure at the 
sea-level. If there were exactly the same quantity of 
air above each square mile of the moon^s surface as 
there is above each square mile of the eartVs surface, 
the lunar air would not only extend to a much greater 
height than ours^ but would be much less dense at 
the moon^s surface. The atmospheric pressure would 
in that case be about l-6th that at our sea-level_, and 
instead of the lower half of such an atmosphere (that 
is, the lower half in actual quantity of air) lying within 
a distance of about 3^ miles from the mean surface, as 
in the case of our earth, it would extend to a distance 
of about 22 miles from the surface. Now this reason- 
ing applies with increased force to the case of an 
atmosphere contained within the cavernous interior of 
the moon, for there the pressure due to the attraction 
of the moon^s mass would be reduced. It is very 
difficult to conceive that under such circumstances 
room would not only exist for lunar oceans, but for a 
lunar atmosphere occupying, one must suppose, a far 
greater amount of space even before its withdrawal 
into these lunar caverns, and partially freed from 
pressure so soon as such withdrawal had taken place. 
That the atmosphere should be withdrawn so com- 
pletely that no trace of its existence could be recog- 
nized, does certainly appear very difficult to believe, 
to say the least. 

Nevertheless, it is not to be forgotten that, so far as 
terrestrial experience is concerned, water is absolutely 
2 A 


essential to tlie occurrence of volcanic action."^ If we 
are to extend terrestrial analogies to tlie case of our 

"" Mr. Mattieu Williams, the author of that valuable and 
suggestive work, "The Fuel of the Sun," makes, however, the 
following remarks on this important subject in an essay in the 
Monthly Notices of the Astronomical Society, which I venture to 
quote nearly in full : 

" Many theoretical efforts, some of considerable violence, have 
been made to reconcile the supposed physical contradiction pre- 
sented by the great magnitude and area of former volcanic activity 
of the moon, and the present absence of water on its surface. So 
long as we accept the generally received belief that water is a ne- 
cessary agent in the evolution of volcanic forces, the difficulties 
presented by the lunar surface are rather increased than diminished 
by further examination and speculation. 

" We know that the lava, scoriae, dust, and other products of 
volcanic action on this earth are mainly composed of mixed sili- 
cates, those of alumina and lime preponderating. When we con- 
sider that the solid crust of the earth is chiefly composed of silicic 
acid, and of basic oxides and carbonates which combine with silicic 
acid when heated, a natural necessity for such a composition of 
volcanic products becomes evident. 

" If the moon is composed of similar materials to those of the 
earth, the fusion of its crust must produce similar compounds, as 
they are formed independently of any atmospheric or aqueous 

" This being the case, the phenomena presented by the cooling 
of fused masses of mixed silicates in the absence of water become 
very interesting. Opportunities of studying such phenomena are 
offered at our great iron-works, where fused masses of iron cinder, 
composed mainly of mixed silicates, are continually to be seen 
in the act of cooling under a variety of circumstances. 

" I have watched the cooling of such masses very frequently, and 
have seen abundant displays of miniature volcanic phenomena, 
especially marked where the cooling has occurred under conditions 
most nearly resembling those of a gradually cooling planet or 


moou^ notwithstanding the signs that the conditions 
prevaiHng in her case have been very different from 

satellite, that is, when the fused cinder has been inclosed by a solid 
resisting and contracting crust. 

" The most remarkable that I have seen are those presented 
by the cooling of the ' tap cinder ' from puddling furnaces. This, 
as it flows from the furnace, is received in stout iron boxes {' cinder 
bogies') of circular or rectangular horizontal section. The follow- 
ing phenomena are usually observable on the cooling of the fused 
cinder in a circular bogie. 

" First a thin solid crust forms on the red-hot surface. This 
speedily cools sufficiently to blacken. If pierced by a shght thrust 
from an iron rod, the red-hot matter within is seen to be in a state 
of seething activity, and a considerable quantity exudes from the 

" If a bogie filled with fused cinder is left undisturbed, a verit- 
able spontaneous volcanic eruption takes place through some por- 
tion, generally near the centre, of the solid crust. In some cases, 
this eruption is sufficiently violent to eject small spirts of molten 
cinder to a height equal to four or five diameters of the whole 

" The crust once broken, a regular crater is rapidly formed, and 
miniature streams of lava continue to pour from it, sometimes 
slowly and regularly, occasionally with jerks and spurts due to the 
bursting of bubbles of gas. The accumulation of these lava-streams 
forms a regular cone, the height of which goes on increasing. I 
have seen a bogie about 10 or 12 inches in diameter, and 9 or 10 
inches deep, thus surmounted by a cone above 5 inches high, with 
a base equal to the whole diameter of the bogie. These cones 
ixnd craters could be but little improved by a modeller desiring to 
represent a typical volcano in miniature. 

" Similar craters and cones are formed on the surface of cinder 
which is not confined by the sides of the bogie. I have seen them 
beautifully displayed on the ' running-out beds ' of refinery furnaces. 
These when filled form a small lake of molten iron covered with a 
layer of cinder. This cinder first skins over, as in the bogies, then 
small crevasses form in this crust, and through these the fused 
2 A 2 


those existing in the case of our earth, we are bound 
to recognize at least the possibility that water once 

cinder oozes from below. The outflow from this chasm soon becomes 
localized so as to form a single crater, or a small chain of craters ; 
these gradually develop into cones by the accumulation of out- 
flowing lava, so that when the whole mass has solidified it is 
covered more or less thickly with a number of such hillocks. 
These, however, are much smaller than in the former case, reaching 
to only one or two inches in height, with a proportionate base. It 
is evident that the dimensions of these miniature volcanoes are 
determined mainly by the depth of the molten matter from which 
they are formed. In the case of the bogies, they are exaggerated 
by the overpowering resistance of the solid iron bottom and sides, 
which force all the exudation in the one direction of least resistance, 
viz., towards the centre of the thin upper crust ;'and thus a single 
crater and a single cone of the large relative dimensions above 
described are commonly formed. The magnitude and perfection of 
these miniature volcanoes vaiy considerably with the quality of 
the pig iron and the treatment it has received, and the difference 
appears to depend upon the evolution of gases, such as carbonic 
oxide, volatile chlorides, fluorides, &c. I mention the fluorides parti- 
cularly, having been recently engaged in making some experiments 
on Mr. Henderson's process for refining pig iron by exposing it when 
fused to the action of a mixture of fluoride of calcium and oxides of 
iron, [alumina, manganese, &c. The cinder separated from this 
iron displayed the phenomena above described very remarkablj^, 
and jets of yellowish flame were thrown up from the craters while 
the lava was flowing. The flame was succeeded by dense white 
vapours as the temperature of the cinder lowered, and a deposit of 
snow-like flocculent crystals was left upon and around the mouth 
or crater of each cone. The miniature representation of conical 
eruptions was thus rendered still more striking, even to the white 
deposit of the haloid salts which Palmieri has described as remain- 
ing after the recent eruption of Vesuvius, 

" The gases thus evolved have not yet been analytically 
examined, and the details of the powerful reactions displayed in 
this process still demand further study, but there can be no doubt 


existed on tlie moon. Moreover, it must be admitted 
that Professor Frankland^s theory seems to accord far 

that the couibinatiou of silicic acid with the base of the fluor spar 
is the fundaineiital reaction to which the evokition of the volatile 
fluorides, &c., is mainly due. 

"A corresponding evolution of gases takes place in cosmical 
volcanic action, whenever silicic acid is fused in contact with 
limestone or other carbonate, and a still closer analogy is presented 
by the fusion of silicates in contact with chlorides and oxides, in 
the absence of water. If the composition of the moon is similar 
to that of the earth, chlorides of sodium, &;c., must form an im- 
portant part of its solid crust ; they should correspond in quantity 
to the great deposit of such salts that would be left behind if the 
ocean of the earth were evaporated to dryness. The only assump- 
tions demanded in applying these facts to the explanation of the 
surface-configuration of the moon are, 1st. That our satellite 
resembles its primary in chemical composition ; 2nd. That it has 
cooled down from a state of fusion ; and, 3rd. That the mag- 
nitude of the eruptions due to such fusion and cooling must bear 
some relation to the quantity of matter in action. 

" The first and second are so commonly made and understood, 
that I need not here repeat the well-known arguments upon which 
they are supported, but may remark that the facts above described 
aftord new and weighty evidence in their favour. 

'' If the correspondence between the form of a freely suspended 
and rotating drop of liquid and that of a planet or satellite is ac- 
cepted as evidence of the exertion of the same forces of cohesion, 
&c., on both, the correspondence between the configuration of the 
lunar surface and that of small quantities of fused and freely cooled 
earth-crust matter, should at least aftord material support to the 
otherwise indicated inference, that the materials of the moons 
crust are similar to those of the earth's, and that they have been 
cooled from a state of fusion. 

" I think I may safely generalize to the extent of saying that no 
considerable mass of fused earthy silicates can cool down under 
circumstances of free radiation, without first forming a heated solid 
crust, which, by further radiation, cooling, and contraction, will 


better with lunar facts than any of the others which 
have been advanced to account for the disappearance 

assume a surface-coniiguration resembling more or less closely that 
of the moon. Evidence of this is afforded by a survey of the 
spoil-banks of blast-furnaces, where thousands of blocks of cinder 
are heaped together, all of which will be found to have their upper 
surfaces (that were freely exposed when cooling) corrugated with 
radiating miniature lava-streams that have flowed from one or more 
craters or openings that have been formed in the manner above 

" The third assumption will, T think, be at once admitted, inas- 
much as it is but the expression of a physical necessity. 

" According to this, the earth, if it has cooled as the moon is 
supposed to have done, should have displayed corresponding irre- 
gularities, and generally, the magnitude of mountains of solidified 
planets and satellites should be on a scale proportionate to their 
whole mass. In comj)aring the mountains of the moon and Mer- 
cury with those of the earth, a large error is commonly made by 
taking the customary measurements of terrestrial mountain-heights 
from the sea-level. As those portions of the earth Avhich rise above 
the waters are but its upper mountain-slopes, and the ocean bottom 
forms its lower plains and valleys, we must add the greatest ocean 
depths to our customary measurements, in order to state the full 
height of what remains of the original mount^iins of the earth. As- 
all the stratified rocks have been formed by the wearing down of 
the original upper slopes and summits, we cannot expect to be able 
to recognize the original skeleton form of our water- washed globe. 

" There is one peculiar feature presented by the cones of the 
cooling cinder which is especially interesting. The flow of fused 
cinder from the little crater is at first copious and continuous, then 
it diminishes and becomes alternating, by a rising and billing of 
the fused mass within the cone. Ultimately the flow ceases, and 
then the inner liquid sinks more or less below the level of the 
orifice. In some cases, where much gas is evolved, this sinking is 
so considerable as to leave the cone as a mere hollow shell, the 
inner liquid having settled down and solidified with a flat or 
slightly rounded surface at about the level of the base of the cone. 


of all traces of water or air. The theory that oceans 
and an atmosphere have been drawn to the farther 

or even lower. These hollow cones were remarkably displayed in 
some of the cinder of the Henderson iron, and their formation was 
obviously promoted by the abundant evolution of gas. 

" If such hollow cones were formed by the cooling of a mass like 
that of the moon, they would ultimately and gradually subside by 
their own weight. But how would they yield ? Obviously by a 
gradual hinge-like bending at the base towards the axis of the 
cone. This would occur with or without fracture, according to the 
degree of viscosity of the crust and the amount of inclination. But 
the sides of the hollow cone-shell, in falling towards the axis, would 
be crushing into smaller circumferences. What would result from 
this ? I thinlc it must be the formation of fissures extending, for 
the most part, radially from the crater towards the base, and a 
crumpling up of the shell of the cone by foldings in the same 
direction. Am I venturing too far in suggesting that in this 
manner may have been formed the mysterious rays and rills that 
extend so abundantly from several of the lunar craters ? 

" The upturned edges or walls of the broken crust, and the 
chasms necessarily gaping between them, appear to satisfy the 
peculiar phenomena of reflection which these rays present. These 
edges of the fractured crust would lean towards each other and 
form angular chasms, while the foldings of the crust itself would 
form long concave troughs extending radially from the crater. 
These, when illuminated by rays falling upon them in the direction 
of the line of vision, would reflect more light towards the spectator 
than would the general convex lunar surface, and thus would 
become especially visible at the full moon. 

" Such foldings and fractures would occur after the subsidence 
and solidification of the lava-forming liquid, that is, when the for- 
mation of new craters had ceased in any given region ; hence they 
would extend across the minor lateral craters formed by outbursts 
from the sides of the main cone, in the manner actually observed. 

" The fact that the bottom of the great walled craters of the moon 
is generally lower than the surrounding plains, must not be forgotten 
in connection with this explanation. 


side of the moon cannot be entertained when due 
account is taken of the range of the lunar librations. 
Sir J. Herschel, indeed, once gave countenance to 
that somewhat bizarre theory ; but he admitted, in a 
letter addressed to myself, that the objection I had 
based on the circumstances of libration was sufficient 
to dispose of the theorj^ The hypothesis that a comet 
had whisked away the lunar oceans and atmosphere 
does not need serious refutation ; and it is difficult to 
see how the theory that lunar seas and lunar air have 
been solidified by intense cold can be maintained in 
presence of the fact that experiments made with the 
Rosse mirror indicate great intensity of heat in the 
substance of those parts of the moon which have been 
exposed to the full heat of the sun during the long 
lunar day. 

If there ever existed a lunar atmosphere and lunar 
seas, then Prof. Frankland's theory seems the only 
available means of accounting for their disappearance. 
Accordingly we must recognize the extreme interest 
and importance of telescopic researches directed to 
the inquiry whether any features of the moon^s surface 
indicate the action of processes of weathering, whether 

" I will not venture further with the speculations suggested by 
the above-described resemblances, as my knowledge of the details 
of the telescopic appearances of the moon is but second-hand. I 
have little doubt, however, that observers who have the privilege 
of direct familiarity with such details, will find that the phenomena 
presented by the cooling of iron cinder or other fused silicates are 
worthy of further and more careful study." 


the beds of lunar rivers can anywhere be traced, 
whether the shores of lunar seas can be recognized by 
any of those features which exist round the coast-lines 
of our own shores. 

One circumstance may be remarked in passing. 
If the multitudinous lunar craters were formed before 
the withdrawal of lunar water and air into the moon^s 
interior, it is somewhat remarkable that the only 
terrestrial features which can be in any way compared 
with them should be found in regions of the earth 
which geologists regard as among those which cer- 
tainly have not been exposed to denudation by the 
action of water. Thus Sir John Herschel, speaking 
of the extinct volcanoes of the Puy de Dome, remarks 
that here the observer sees " a magnificent series of 
volcanic cones, fields of ashes, streams of lava, and 
basaltic terraces or platforms, proving the volcanic 
action to have been continued for countless ages 
before the present surface of the earth was formed ; 
here can be seen a configuration of surface quite 
resembling what telescopes show in the most volcanic 
districts of the moon ; for half the moon^s face is 
covered with unmistakable craters of extinct vol- 
canoes.^^ But Lyell, speaking of the same volcanic 
chains, describes them as regions " where the eruption 
of volcanic matter has taken place in the open air, and 
where the surface has never since been subjected tc 
great aqueous denudation.^' If all the craters on the 
moon belonged to one epoch, or even to one era, we 
might regard them as produced during the withdrawal 


of the lunar oceans within the still heated substance 
of our satellite. But it is manifest that the processes 
which brought the moon^s surface to its present con- 
dition must have occupied many ages^ during which 
the craters formed earliest would be exposed to the 
effects of denudation_, and to other processes of which 
no traces can be recognized. It is not likely, how- 
ever, that the withdrawal of the lunar oceans into the 
moon^s cavernous interior can have taken place sud- 
denly. Up to a certain epoch the entry of the waters 
within the moon^s mass would be impossible, owing 
to the intense heat, which, by maintaining the plas- 
ticity of the moon^s substance, would prevent the 
formation of cavities and fissures, while any water 
brought into contact with the heated interior would at 
once be vaporized, and driven away. But when once 
a condition was attained which rendered the formation 
of cavities possible, the contraction of the moon^s sub- 
stance would lead to the gradual increase of such 
cavities, and so, as time proceeded, room would be 
found for all the lunar oceans. 

We are next led to the inquiry whether the contrac- 
tion of the moon^s substance may not have played the 
most important part of all in producing those pheno- 
mena of disturbance which are presented by the 
moon^s surface. Quite recently the eminent seis- 
mologist Mallet has propounded a theory of terrestrial 
volcanic energy, which not only appears to account 
far more satisfactorily than any hitherto adopted for 
the phenomena presented by the eartVs crust, but 


suggests considerations wMcli may be applied to the 
case of the nioon_, and in fact are so applied by Mallet 
himself. It behoves us to inquire very carefully into 
the bearing of this theory upon the subject of lunar 
seismology^ and therefore to consider attentively the 
points in which the theory differs from those hitherto 

Mallet dismisses first the chemical theory of volcanic 
energy^ because all known facts tend to show that the 
chemical energies of the materials of our globe were 
almost wholly exhausted prior to the consolidation of 
its surface. This may be regarded as equally appli- 
cable to the case of the moon. It is difficult to see 
how the surface of the moon can have become con- 
solidated while any considerable portion of the che- 
mical activity of her materials remained unexhausted. 

" The mechanical theory/^ proceeds Mallet^ 'Svhich 
finds in a nucleus still in a state of liquid fusion a 
store of heat and of lava^ &c._, is only tenable on the 
admission of a very thin solid crust ; and even through 
a crust but 30 miles thick_, it is difficult to see how 
surface-water is to gain access to the fused nucleus ; 
yet witlwid water there can he no volcano. More recent 
investigation on the part of mathematicians has been 
supposed to prove that the eartVs crust is not thin." 
He proceeds to show that, without attaching any great 
weight to these mathematical calculations, there are 
other grounds for believing that the solid crust of the 
earth is of great thickness, and that " although there 
is evidence of a nucleus much hotter than the crust, 


there is no certainty that any part of it remains liquid; 
but if so_, it is in any case too deep to render it con- 
ceivable that surface-water should make its way down 
to it. The results of geological speculation and of 
physico-mathematical reasoning thus oppose each 
other; so that some source of volcanic heat closer to 
the surface remains to be sought. The hypothesis to 
supply this, proposed by Hopkins and adopted by 
some, viz., of isolated subterranean lakes of liquid 
matter, in fusion at no great depth from the surface, 
remaining fused for ages, surrounded by colder and 
solid rock, and with (by hypothesis) access of surface- 
water, seems feeble and unsustainable.'^ 

Now in some respects this reasoning is not ap- 
plicable to the moon, at least so far as real evidence is 
concerned; though it is to be noticed that, if a case 
is made out for any cause of volcanic action on the 
earth, we are led by analogy to extend the reasoning 
(or at least its result) to the case of the moon. But 
it may be remarked that the solidification of the 
moon's crust must have proceeded at a more rapid 
rate than that of the earth's, while the proportion of 
its thickness to the volume of the fused nucleus would 
necessarily be greater for the same thickness of the 
crust. The question of the access of water brings us 
to the difficulty already considered, the inquiry, 
namely, whether oceans originally existed on the 
moon. For the moment, however, we forbear from 
considering whether Mallet's reasoning must neces- 
sarily be regarded as inapplicable to the moon if it 


should be admitted that there never were any lunar 

"We come now to Mallet's solution of the problem of 
terrestrial volcanic energy. 

We have been so long in the habit of regarding 
volcanoes and earthquakes as evidences of the earth's 
subterranean forces_, as due^ in fact (to use Hum- 
boldt's expression), to the reaction ofthe earth's interior 
upon its crust, that the idea presents itself at first 
sight as somewhat startling, that all volcanic and 
seismic phenomena, as well as the formation of moun- 
tain-ranges, have been due to a set of cosmical forces 
called into play by the contraction of our globe. Ac- 
cording to the new theory, it is not the pressure of 
matter under the crust outwards, but the pressure of 
the earth's crust inwards, which produces volcanic 
energy. Nor is this merely substituting an action 
for reaction, or vice versa. According to former 
views, it was the inability of the crust to resist 
pressure from within which led to volcanic explosions, 
or which produced earthquake-throes where the 
safety-valve provided by volcanoes was not supplied. 
The new theory teaches, in fact, that it is a deficiency 
of internal resistance, and not an excess, which causes 
these disturbances of the crust. ^'The contraction 
of our globe," says Mallet,* " has been met, from the 

* I quote throughout from an abstract of Mallet's paper in the 
Fhilosojphical Magazine for December, 1872. The words are pro- 
bably, for the most part. Mallet's own ; but I have not the original 
paper by me for reference. I believe, however, that the abstract is 
from his own pen. 


period of its fluidity to its present state, first, by 
deformation of the spheroid, forming generally the 
ocean-basins and the land ; afterwards by the foldings 
over and elevations of the thickened crust into moun- 
tain-ranges, &c.; and, lastly, by the mechanism which 
gives rise to volcanic actions. The theory of moun- 
tain-elevation proposed by C. Provost was the only 
true one,' that which ascribes this to tangential 
pressures propagated through a solid crust of sufficient 
thickness to transmit them, these pressures being 
produced by the relative rate of contraction of the 
nucleus and of the crust ; the former being at a higher 
temperature, and having a higher coefficient of con- 
traction for equal loss of heat, tends to shrink away 
from beneath the crust, leaving the latter partially 
unsupported. This, which during a much more rapid 
rate of cooling from higher temperature of the whole 
globe, and from a thinner crust, gave rise in former 
epochs to mountain- elevation, in the present state 
of things gives rise to volcanic heat.^^ By the ap- 
plication of a theorem of Lagrange, Mr. Mallet proves 
that the earth^s solid crust, however great may be its 
thickness, ^'^and even if of materials far more co- 
hesive and rigid than those of which we must suppose 
it to consist, must, if even to a very small extent left 
unsupported by the shrinking away of the nucleus, 
crush up in places by its own gravity, and by the 
attraction of the nucleus. This is actually going on; 
and in this partial crushing," at places or depths 
dependent on the material and on conditions which 


Mr. Mallet points out, he discerns '^ tlie true cause of 
volcanic heat. * As the solid crust sinks together to 
follow down after the shrinking nucleus^ the ivorh 
expended in mutual crushing and dislocation of its 
parts is transformed into lieatj by which^ at the places 
where the crushing sufficiently takes place^ the ma- 

* " In order to test the validity of his theory by contact with 
known facts " (says the Philosojjhical Maganne), " Mr. Mallet gives 
in detail two important series of experiments completed by him ; 
the one on the actual amount of heat capable of being developed 
by the crushing of sixteen different species of rocks, chosen so as to 
be representative of the whole series of known rock-formations, 
from oolites down to the hardest crystalline rocks ; the other on 
the coefficients of total contraction between fusion and sohdification, 
at existing mean temperature of the atmosphere, of basic and acid 
slags analogous to melted rocks. The latter experiments were 
conducted on a very large scale ; and the author points out the 
great errors of preceding experimenters, Bischoff and others, as to 
these coefficients. By the aid of these experimental data, he is 
enabled to test the theory produced when compared with such 
facts as we possess as to the rate of present cooling of our globe, 
and the total annual amount of volcanic action taking place upon 
its surface and within its crust. He shows, by estimates which 
allow an ample margin to the best data we possess as to the total 
annual vulcanicity, of all sorts, of our globe at present, that less 
than one-fourth of the total heat at present annually lost by our 
globe is upon his theory sufficient to account for it ; so that the 
secular cooling, small as it is, now going on, is a sufficient ijrimum 
mobile, leaving the greater portion still to be dissipated by radia- 
tion. The author then brings his views into contact with known 
facts of vulcanology and seismology, showing their accordance. He 
also shows that to the heat developed by partial tangential thrusts 
within the solid crust are due those perturbations of hypogeal 
increment of temperature which Hopkins has shown cannot be 
referred to a cooling nucleus and to differences of conductivity 


terial of the rock so crushed and of that adjacent to 
it are heated even to fusion. The access of water to 
such points determines volcanic eruption. Volcanic 
heat^ therefore, is one result of the secular cooling of 
a terraqueous globe subject to gravitation, and needs 
no strange or gratuitous hypothesis as to its origin." 

It is readily seen how important a bearing these 
conclusions have upon the question of the moon's 
condition. So far, at any rate, as the processes of 
contraction and the consequent crushing and dis- 
location of the crust are concerned, we see at once 
that in the case of the moon these processes would 
take place far more actively than in the earth's case. 
For the cooling of the moon must have taken place 
far more rapidly, and the excess of the contraction of 
the nucleus over that of the crust must have been 
considerably greater. Moreover, although the force 
of gravity is much less on the moon than on our earth, 
and therefore the heat developed by any process of 
contraction correspondingly reduced, yet, on the one 
hand, this would probably be more than compensated 
by the greater activity of the lunar contraction (i.e, 
by the more rapid reduction of the moon's heat), and 
on the other, the resistance to be encountered in the 
formation of elevations by this process would be re- 
duced precisely in the same proportion that gravity 
is less at the moon's surface. It is important to 
notice that, as Mr. Mallet himself points out, his view 
of the origin of volcanic heat " is independent of any 
particular thickness being assigned to the earth's 


solid crusty or to whether there is at present a liquid 
fused nucleus, all that is necessary being a hotter 
nucleus than crust, so that the rate of contraction is 
greater for the former than for the latter/' Moreover, 
" as the play of tangential pressures has elevated the 
mountain-chains in past epochs, the nature of the 
forces employed sets a limit '' to the possible height 
of mountains on our globe. This brings Mr. Mallet's 
views into connection with " vulcanicity produced in 
like manner in other planets, or in our own satellite, 
and supplies an adequate solution of the singular, and 
so far unexplained fact, that the elevations upon our 
moon's surface and the evidences of former volcanic 
activity are upon a scale so vast when compared with 
those upon our globe.'' 

All that seems wanted to make the explanation of 
the general condition of the moon's surface complete, 
according to this theory, is the presence of water in 
former ages, over a large extent of the moon's 
surface, unless we combine with the theory of con- 
traction the further supposition that the downfall of 
large masses on the moon produced that local fusion 
which is necessary to account for the crateriform 
surface-contour. It is impossible to contemplate the 
great mountain-ranges of the moon (as, for instance, 
the Apennines under favourable circumstances of 
illumination) without seeing that Mallet's theory 
accords perfectly with their peculiar corrugated 
aspect (the same aspect, doubtless, which terrestrial 
mountain-ranges would exhibit if they could be 
2 b 


viewed as a wliole from any suitable station). Again, 
the aspect of the regions surrounding the great lunar 
craters and especially the well-studied crater Coper- 
nicus accords closely,, when sufficient telescopic power 
is employed, with the theory that there has been a 
general contraction of the outer crust of the moon, 
resulting in foldings and cross-foldings, wrinkles, 
corrugations, and nodules. But the multiplicity of 
smaller craters does not seem to be explained at all 
satisfactorily ; while the present absence of water, as 
well as the want of any positive or direct evidence 
that water ever existed upon the moon, compels us to 
regard even the general condition of the moon^s 
surface as a problem which has still to be explained. 
If, however, it be admitted that the processes of 
contraction proceeded with sufficient activity to pro- 
duce fusion in the central part of a great region of 
contracting crust, and that the heat under the crust 
sufficed for the vaporization of a considerable portion 
of the underlying parts of the moon^s substance, we 
might find an explanation of the great craters like 
Copernicus, as caused by true volcanic action. The 
masses of vapour which, according to that view, sought 
an outlet at craters like Copernicus must have been 
enormous however. Almost immediately after their 
escape they would be liquefied, and flow down outside 
the raised mouth of the crater. According to this 
view we should see, in the floor of the crater, the 
surface of what had formerly been the glowing nucleus 
of the moon : the masses near the centre of the floor 


(in so many cases) might be regarded as, in some 
instances, the debris left after the great outburst, and 
in others as the signs of a fresh outburst proceeding 
from a yet lower level ; while the glistening matter 
which lies all round many of the monster craters 
would be regarded as the matter which had been 
poured out during the outburst. 

We need not discuss in this connection the minor 
phenomena of the moon^s surface. It seems evident 
that the rilles, and all forms o^ faults observable on the 
moon's surface, might be expected to result from such 
processes of contraction as Mallet's theory deals with. 

It is, in fact, the striking features of the moon's 
disc those which are seen when she is examined with 
comparatively low telescopic powers which seem to 
tax most severely every theory which has yet been 
presented. The clustering craters, which were com- 
pared by Galileo to ^' eyes upon the peacock's tail," 
remain unaccounted for hitherto ; and so do the great 
dark regions called seas. Mallet's theory explains^ 
perhaps, the varieties of level observed in the moon's 
surface-contour, but the varieties of tint and colour 
remain seemingly inexplicable. 

There is one feature of the lunar globe which pre- 
sents itself to us under a wholly changed aspect if 
we adopt Mallet's theory. I refer to the radiations 
described at pp. 251-2. According to any theory which 
accounted for these features as due to internal forces 
acting outwards, it was exceedingly difficult to inter- 
pret the fact that along the whole length of these rays 
2 B 2 


there can be observed a peculiar difference of bright- 
ness under direct illumination, while, nevertheless, 
such features of the surface as craters, mountain- 
ranges, plains, and so on, extend unbroken over the 
rays. I do not know that the theory of contraction 
serves to meet the difficulty completely ; in fact, the 
difference of tint in the rays, and the circumstance that 
the rays can ouly be well seen under full illumination, 
appear to me to be among the most perplexing of the 
many perplexing phenomena presented by the moon's 
surface. But so far as the mere formation of radiations 
of enormous length is concerned, it seems to me that 
we have a far more promising interpretation in the 
theory of contraction than in any theory depending on 
the action of sublunarian forces. For whenever an 
outer crust is forced to contract upon an enclosed 
nucleus, a tendency can be recognized to the formation 
of radially-arranged corrugations. Nevertheless, it 
may be questioned whether when this tendency is 
most clearly recognized there is not always present 
some unyielding matter which forms a centre round 
which the radiations are formed; and it is somewhat 
difficult to see how or why such centres of resistance 
should exist in the case of the lunar crust. It is a 
little remarkable that here again we find ourselves led 
to entertain the notion that matter arriving from with- 
out has produced these sublunarian hiots, if one may 
so speak, whose presence is not directly discernible, 
but is nevertheless strikingly indicated by these series 
of radiating streaks. 


The circumstance already referred to^ that these 
rays can only be well seen when the moon is full, has 
long and justly been regarded as among the most mys- 
terious facts known respecting the moon. It is difficult 
to understand how the peculiarity is to be explained 
as due merely to a difference of surface-contour in the 
streaks; for it is as perplexing to understand how the 
neighbouring regions could darken from this cause 
just before full moon_, and remain relatively dark 
during two or three days^ as to explain the peculiarity 
by supposing that the rays themselves grow relatively 
bright. It is true that there are certain surfaces which 
appear less bright under a full than under an oblique 
illumination^ using the words ' full ^ and ' oblique ' 
with reference to the general level of the surface. But 
the radiations occupy arcs of such enormous length 
upon the moon^s surface, that the actual illumination 
of different parts of the radiations varies greatly, and 
of course there is a like variation in the illumination 
of different parts of the regions adjacent. 

It is natural, under these circumstances, to inquire 
how far it is probable (1) that real processes of change 
take place month by month on the moon's surface, 
and (2) that it is to these processes that we owe the 
greater or lesser distinctness with which certain 
features present themselves. 

We have seen that Dr. De la Rue was led, by his 
photographic researches into the moon's condition 
(for we may fairly thus describe his experience in 
lunar photography), to the conclusion that processes 


resembling vegetation take place on the moon^ the 
period during which the vegetation passes through 
its series of changes being a lunar month^ and that 
the moon may have an atmosphere of great density, 
but of small extent. 

It is extremely important to notice that photo- 
graphy shows the light near the terminator to be 
less bright than it appears to the eye. It may be, of 
course, that the distinction resides mainly or entirely 
between the photographic power and the luminosity 
of these portions ; there may, for example, be an 
excess of yellow light and a deficiency of green, 
while the greater photographic power of the parts 
under full solar illumination may indicate an increase 
of green light due to some process of vegetation. It 
is, however, important to inquire whether the greater 
part of the difference may not be due to a physio- 
logical cause ; whether, in fact, the neighbourhood of 
the dark portion of the disc may not cause the illumi- 
nated parts near the terminator to appear, through 
contrast, brighter than they really are. 

On the answer which may be given to this question 
depends, in a great degree (as it seems to me), the 
opinion we are to form of those recent researches by 
Mr. Birt which have appeared to indicate that the floor 
of Plato grows darker as the sun rises higher above it. 
Taking these researches in their general aspect, it 
cannot but be recognized that :t is a matter of tho 
utmost importance to determine whether they indi- 
cate a real change or one which is only apparent. If 


it is really the case that Plato grows darker under a 
rising sun, we should have to infer that in the case of 
Plato certainly, and probably in the case of other 
regions similarly placed, processes of change take 
place in each lunation which correspond (fairly) with 
what might be expected if these regions became 
covered with some sort of vegetation as the lunai 
month (or, which is the same thing, the lunar day) 
.proceeds. Other explanations meteorological, che- 
mical, or mechanical might indeed be available, yet 
in any case conclusions of the utmost interest would 
present themselves for consideration. 

It must be remembered, however, that thus far 
Mr. Births observations (as well those made by 
himself as those which he has collected together) 
are based on eye- estimations. Nothing has yet been 
done to apply any photometric test to the matter; 
nor has the floor of Plato been brotight alone under 
observation, but other light, of varying degrees of 
intensity, has always been in the field of view. Plato 
is seen bright when near the 'terminator,^ and 
growing gradually darker as the sun rises higher 
and higher above the level of the floor of the crater. 
The point to be decided is, how far the brightness of 
Plato near the terminator is an effect of contrast. De 
la Pue^s photographic observations go far to prove 
(they at least strongly suggest) that contrast has 
much to do with the matter. He has shown that, 
photographically, the parts near the terminator are 
not so bright as they look. May it not be that they 


look brighter than they are in reality? We have 
only to suppose that De la Kue^s photographic results 
represent pretty accurately the true relative luminosity 
of different parts of the moon to answer this question 
at once in the aflSrmative. 

It seems to accord with this view_, that the greater 
darkness of the floor of Plato agrees,, according to Mr. 
Births light-curveSj with the time when the sun attains 
his greatest elevation above the level of the floor. 
For if the action of the sun vvere the cause of the 
darkening, we should expect the greatest efiect to 
appear some considerable time after the sun had 
culminated (as supposed to be seen from the floor of 
Plato). We know that on our own earth all diurnal 
solar effects, except those which may be described 
as optical, attain their maximum after the sun has 
reached his highest point on the heavens, while all 
annual solar effects attain their maximum after 
midsummer. If an observer on Yenus could watch 
the forests of our north temperate zones as 
they became clothed of vegetation, and were after- 
wards disrobed of their leafy garment during the 
progress of the year, it would not be on the 21st of 
June that he would recognize the most abundant 
signs of vegetation. In July and August vegetation 
most richly clothes the northern lands of our earth. 
It is then also that the heat is greatest ; that is the 
time of true midsummer as distinguished from astro- 
nomical midsummer. And in like manner the true 
heat- noon is at about two o^clock in the afternoon. 


not at the epocli when the sun is highest, or at astro- 
nomical noon. The difference in either case amounts 
to about one-twelfth part of the complete period in 
question : in one case we find the maximum of heat a 
month or twelfth part of the year after the time of the 
sun^s greatest northerly declination ; in the other we 
find the time of greatest heat two hours, or one- 
twelfth part of a day, after the time of the sun^s 
greatest elevation. If we take a corresponding por- 
tion of the lunar month, we find that the greatest 
effect of any solar action on the floor of Plato might 
be expected to take place about two and a half days 
after the sun had attained his greatest elevation. 
This differs to a sufficient degree from Mr. Births 
estimate to justify the suspicion that either the effect 
is physiological, or that it is purely an optical pecu- 
liarity, that is, due to the manner in which the light 
falls on a surface of peculiar configuration. 

It does not appear to me, I may remark further, 
that Mr. Birt has demonstrated the occurrence of real 
variations in the condition of the spots upon the floor 
of Plato. He has ascertained that some of these are 
at times relatively darker or brighter than at others, 
and that this is not a mere physiological effect is 
proved by the fact that the result has been obtained 
by comparing the spots inter se. Nevertheless it 
must not be forgotten how largely the presentation of 
the floor of Plato towards the terrestrial observer is 
affected by libration, now tilting the floor more fully 
towards the observer and presently tilting it away 


from him; at one time tilting the floor eastwards, 
at another westwards, and at intermediate periods giv- 
ing every intermediate variety of tilt; these changes, 
moreover, having their maximum in turn at all epochs 
of the lunation. Combining this consideration with 
the circumstance that very slight variations in the 
presentation of a flattish surface will cause certain 
portions to appear relatively dark or relatively light, 
it appears to me that a case has not yet been made 
out for those selenographical changes by which Mr. 
Birt has proposed to interpret these phenomena. 

Nevertheless it cannot be insisted on too strongly 
that it is from the detailed examination of the moon^s 
surface that we can now alone hope for exact informa- 
tion as to its present condition and past history. I 
would even urge, indeed, that the detailed examina- 
tion at present being carried out is not sufficiently 
exact in method. I should be glad to hear of such 
processes of examination as were applied by Mr. 
Dawes to the solar spots. In particular it seems to 
me most important that the physiological effects 
which render ordinary telescopic observation and 
ordinary eye-estimates of size, brightness, and colour 
deceptive, should be as far as possible eliminated. 
This might be done by so arranging the observations 
that the conditions under which each part of the moon 
should be studied might be as far as possible equalized 
during the whole progress of the lunation. Thus, 
returning to the case of the floor of Plato : this region 
should not be examined when Plato is near the ter- 


minator as well as at the time of full moon, with the 
rest of the moon's disc or large portions thereof in the 
field of view ; the eye of the observer should be pro- 
tected from all light save that which comes from the 
floor itself; and, moreover, the artificial darkness 
produced for this purpose should be so obtained that 
the general light of the full moonlight should be 
excluded as well as the direct light from the disc. 
Then difierences of tint should be carefully estimated 
either by means of graduated darkening-glasses, or 
by the introduction of artificially illuminated surfaces 
into the field of view for direct comparison with the 
lunar region whose brightness is to be determined. 

When observations thus carefully conducted are 
made, and when the efi'ects of libration as well as of 
the sun's altitude above the lunar regions studied are 
carefully taken into account, we should be better able 
than we are at present, as it appears to me, to deter- 
mine whether the moon's surface is still undergoing 
changes of configuration. I cannot but think that 
such an inquiry would be made under more promising 
circumstances than those imagine who consider that 
the moon's surface has reached its ultimate condition, 
and that therefore the search for signs of change is a 
hopeless one. So far am I from considering it unlikely 
that the moon's surface is still undergoing change, 
that, on the contrary, it appears to me certain that 
the face of the moon must be undergoing changes of 
a somewhat remarkable nature, though not producing 
any results which are readily discerned by our imperfect 


telescopic means. It is not difficult to show reasons 
at least for believing that the face of the moon must 
be changing more rapidly than that of our earth. On 
the earth, indeed, we have active subterranean forces 
which may, perhaps, be wanting in the moon. On the 
earth again, we have a sea acting constantly upon the 
shore, here removing great masses, there using the 
dthris to beat down other parts of the coast, and by 
the mere effect of accumulated land-spoils acquiring 
power for fresh inroads. We have, moreover, wind 
and rain, river action, and glacier action, and, lastly, 
the work of living creatures by land and by sea, while 
most of these causes of change may be regarded as 
probably, and some as certainly, wanting in the case 
of our satellite. Nevertheless, there are processes at 
work out yonder which must be as active, one cannot 
but believe, as any of those which affect our earth. 
In each lunation, the moon^s surface undergoes changes 
of temperature which should suffice to disintegrate 
large portions of her surface, and with time to crumble 
her loftiest mountains into shapeless heaps. * In the 
long lunar night of fourteen days, a cold far exceed- 
ing the intensest ever produced in terrestrial ex- 
periments must exist over the whole of the unillu- 
minated hemisphere ; and under the influence of this 
cold all the substances composing the moon^s crust 
must shrink to their least dimensions not all equally 
(in this we find a circumstance increasing the energy 

* Nasmyth pointed this out long since. 


of the disintegrating forces), but eacli according to 
the quality which our physicists denominate the 
coefficient of expansion. Then comes on the long 
lunar day, at first dissipating the intense cold, then 
gradually raising the substance of the lunar crust to 
a higher and higher degree of heat, until (if the in- 
ferences of our most skilful physicists, and the evidence 
obtained from our most powerful means of experiment 
can be trusted) the surface of the moon burns (one 
may almost say) with a heat of some SOO*' F. Under 
this tremendous heat all the substances which had 
shrunk to their least dimensions must expand accord- 
ing to their various degrees ; not greatly, indeed, so 
far as any small quantity of matter is affected, but to 
an important amount when large areas of the moon's 
surface are considered. Remembering the effects 
which take place on our earth, in the mere change 
from the frost of winter to the moderate warmth of 
early spring, it is difficult to conceive that such 
remarkable contraction and expansion can take place 
in a surface presumably less coherent than the re- 
latively moist and plastic substances comprising the 
terrestrial crust, without gradually effecting the 
demolition of the steeper lunar elevations. When we 
consider, further, that these processes are repeated 
not year by year, but month by month, and that all 
the circumstances attending them are calculated to 
render them most effective because so slow, steadfast, 
and uniform in their progression, it certainly does not 
seem wonderful that our telescopists should from time 


to time recognize signs of cliange in the moon^s face. 
So far from rejecting tliese as incredible^ we should 
consider the wonder rather to be that they are not 
more commonly seen, and more striking in their 
nature. Assuredly there is nothing which should lead 
our telescopists to turn from the study of the moon, as 
though it were hopeless to seek for signs of change 
on a surface so desolate. Eather they should increase 
the care with which they pursue their observations, 
holding confidently the assurance that there are signs 
of change to be detected, and that in all probability 
the recognition of such change may throw an instruc- 
tive light on the moon's present condition, past history, 
and probable future. 




Grey Plains, usually called Seas. 

A. Mare Crisium 

B. Humboldti- 


C. Frigoris 

D. Lacus Mortis 

E. Somuiorum 

F. Palus Somnii 

G. Mare Tranquilli- 


H. Mare Serenitatis 
I. Palus Nebula- 

K. Putredinis 

L. Mare Vaporum 
M. Sinus Medii 
N. * ^Estuum 
0. Mare Imbrium 
P, Sinus Iridum 

d, Oceanus Procel- 

K. Sinus Eoris 
S. Mare Nubiuni 

T. Humoruni 

V. Nectaris 

X. Foecundi- 

Z. Australe 


Craters, Mountains, and other Objects. 
Numbered as in the Maj). 

1. Promontorium 


2. Alhazen 

3. Einimart 

4. Picard 

5. Condorcet 

6. Azout 

7. Firmicus 

8. ApoUonius 

9. Neper 

10. Schubert 

11. Hansen 

12. Cleomedes 

13. Tralles 

14. Oriani 

15. Plutarchus 

16. Seneca 

17. Hahn 

18. Berosus 

19. Burckhardt 

20. Geminus 



21. Bernouilli 

22. Gauss 

23. Messala 

24. Schumacher 

25. Struve 

26. Mercurius 

27. Endymion 

28. Atlas 

29. Hercules 

30. Oersted 

31. Cepheus 

32. Franklin 

33. Berzelius 

34. Hooke 

35. Strabo 

36. Thales 

37. Gartner 

38. Democritus 

39. Arnold 

40. Christian Mayer 

41. Meton 

42. EuctenJon 

43. Scoresby 

44. Gioja 

45. Barrow 

46. Archytas 

47. Plana 

48. Mason 

49. Baily 

50. Burg 

51. Mt. Taurus 

52. Romer 

53. Le Monnier 

54. Posidonius 

55. Littrrow 

56. Maraldi 

57. Vitruvius 

58. [Mt. Argseus] 

59. Macrobius 

60. Proclus 

61. Plinius 

62. Ross 

63. Arago 

64. Ritter 

65. Sabine 

66. Jansen 

67. Maskelyne 

68. Mt. Hsemus 

69. Promontorium. 


70. Menelaus 

71. Sulpicius Gallus 

72. Taquet 

73. Bessel 

74. Linne 

75. Mt. Caucasus 

76. Calippus 

77. Eudoxus 

78. Aristoteles* 

79. Eged 

80. Alps 

81. Cassini 

82. Thecetetus 

83. Aristillus 

84. Autolycus 

85. Apennines 

86. Aratus 

87. Mt. Hadley 

88. Conon 

89. Mt. Bradley 

90. Mt. Huygens 

91. Marco Polo 

92. Mt. Wolf 

93. Hyginus 

94. Triesneckner 

95. Manilius 

96. Julius Cffisar 

97. Sosigenes 

98. Boscovich 

99. Dionysius 

100. Ariadajus 

101. Silberschlag 
102; Agrippa 

103. Godin 

104. Rhseticus 

105. Sommering 

106. Schrbter 

107. Bode 

108. Pallas 

109. Ukert 

110. Eratosthenes 

111. Stadius 

112. Copernicus 

113. Gambart 

114. Reinhold 

115. Mt. Carpathus 

116. Gay-Lussac 

117. Toiiias Mayer 

118. Milichius 

119. Hortensius 

120. Archimedes 

121. Timocharis 

122. Lambert 

123. La Hire 

124. Pytheas 

125. Euler 

126. Diophantus 

127. Delisle 

128. Carlini 

129. Helicon 

130. Kirch 

131. Pico 

132. Plato 

133. Harpalus 



134. Laplace 172. 

135. Heraclides 173. 

136. Maupertuis 174. 

137. Condamine 175. 

138. Bianchini 176. 

139. Sharp 177. 

140. Mairan 178. 

141. Louville 179. 

142. Bouguer 180. 

143. Encke 181. 

144. Kepler 182. 

145. Bessarion 183. 

146. Eeiner 184. 

147. Marius 185. 

148. Aristarchus 186. 

149. Herodotus 187. 

150. Wollaston 188. 

151. Lichtenberg 189. 

152. Harding 190. 

153. Lohrmann 191. 

154. Hevel 192. 

155. Cavalerius 193. 

156. Galileo 194. 

157. Cardanus 195. 

158. Krafft 196. 

159. Olbers 197. 

160. Vasco de Gama 198. 

161. Hercynian Mts. 199. 

162. Seleucus 200. 

163. Briggs 201. 

164. Ulugh Beigh 202. 

165. Lavoisier 203. 

166. Gerard 204. 

167. Kepsold 205. 

168. Anaxagoras 206. 

169. Epigenes 

170. Tim^us 207. 

171. Fontenelle 208. 



, Horrebow 
















Wilheltn I. 

Ebngomon tanu s 
















2 c 

209. Davy 

210. Lalande 

211. Mosting 

212. Herschel 

213. Bullialdus 

214. Kies 

215. Guerike 

216. Lubiniezky 

217. Parry 

218. Bonpland 

219. Era Mauro 

220. RiphseanMts. 

221. Euclides 

222. Landsberg 

223. Flamsteed 

224. Letronne 

225. Hippalus 

226. Campanus 

227. Mercator 

228. Ramsden 

229. Vitello 

230. Doppelmayer 

231. Mersenius 

232. Gassendi 

233. Agatharchides 

234. Schiller 

235. Bayer 

236. Rost 

237. Hainzel 

238. Capuanus 

239. Schickard 

240. Drebbel 

241. Lehmann 

242. Phocylides 

243. Wargentin 

244. Inghiraini 

245. Bailly 

246. Dorfel Mts. 



247. Hausen 285. Fourier 

248. Segner 286. Cavendish 

249. Weigel 287. Reaumur 

250. Zuchius 288. Hipparchus 

251. Bettinus 289. Albategnius 

252. Kircher 290. Parrot 

253. Wilson 291. Airy 

254. Casatus 292. La Caille 

255. Klaproth 293. Playfair 

256. Newton 294. Apianus 

257. Cabeus 295. Werner 

258. Malapert 296. Aliacensis 

259. Leibnitz Mts. 297. Theon, sen. 

260. Blancanus 298. Theon, jun. 

261. Scheiner 299. Taylor 

262. Moretus 300. Alfraganus 

263. Short 301. Delambre 

264. Cysatus 302. Kant 

265. Gruemberger 303. Dollond 

266. Billy 304. Descartes 

267. Hansteen 305. Abulfeda 

268. Zupus 306. Almanon 

269. Fontana 307. Tacitus 

270. Sirsalis 308. Geber 

271. Damoiseau 309. Azophi 

272. Grimaldi 310. Abenezra 

273. Riccioli 311. Pontanus 

274. Cordilleras 312. Sacrobosco 
275.DAlerabertMts. 313. Pons 

276. Rook Mts. 314. Fermat 

277. Rocca 315. Altai Mts. 

278. Criiger 316. Polybius 

279. Byrgius 317. Hypatia 

280. Eichstadt 318. Torricelli 

281. Lagrange 319. Theophilus 

282. Piazzi 320. Cyrillus 

283. Bouvard 321. Catharina 
234 Vieta 322. Beaumont 

323. Isidorus 

324. Capella 

325. Censorinus 

326. Taruntius 

327. Messier 

328. Goclenius 

329. Biot 

330. Guttemberg 

331. Pyrenees 

332. Bohnenberger 

333. Colombo 

334. Magelhaens 

335. Cook 

336. Santbech 

337. Borda 

338. Langrenus 

339. Vendelinus 

340. Petavius 

341. Palitzsch 

342. Hase 

343. Snellius 

344. Stevinus 

345. Furnerius 

346. Maclaurin 

347. Kastner 

348. Lapeyrouse 

349. Ansgarius 

350. Behaini 

351. Hecatseus 

352. Wilhelm Hum- 


353. Legendre 

354. Stofler 

355. Licetus 
:356. Cuvier. 

357. Clairaut 

358. Maurolycus 

359. Barocius 



360. Bacon 

361. Buch 

362. Biisching 

363. Gemma Frisius 

364. Poisson 

365. Nonius 

366. Fernelius 

367. Riccius 

368. Rabbi Levi 

369. Zagut 

370. Lindenau 

371. Piccolomini 

372. Fracastoriiis 

373. Neander 

374. Stiborius 

375. Reicbenbach 

376. Rbeita 

377. Fraunhofer 

378. Vega 

379. Marinus 

380. Oken 

381. Pontecoulant 

382. Hanuo 

383. Fabricius 

384. Metius 

385. Steinheil 

386. Pitiscus 

387. Hommel 

388. Vlacq 

389. Rosenberger 

390. Nearchus 

391. Hagecius 

392. Biela 

393. Nicolai 

394. Lilius 

395. Jacobi 

396. Zach 

397. Schomberger 

398. Boguslawsky 435. 

399. Boussingault 436. 

400. Mutus 437. 

401. Manzinus 438. 

402. Pentland 

403. Simpelius 439. 

404. Curtius 440. 

405. Coxwell Mts. 441. 

406. Mt. Glaisher 442. 

407. Chevallier 443. 

408. Moigno 444. 

409. Peters 445. 

410. Tenerife Mts. 446. 

411. Smyth, Piazzi 447. 

412. Herschel,J.F.W.448. 

413. Robinson 

414. South 449. 

415. Babbage 450. 

416. Percy Mts. 

417. Rosse 451. 

418. Franklin, J. 

419. Crozier 452. 

420. McClure 

421. Bellot 453. 

422. Wrottesley 454. 

423. Phillips 455. 

424. Mare Smythii 456. 

425. Le Verrier 457. 

426. Shuckburgh 458. 

427. Goldschmidt 459. 

428. Rumker 460. 

429. Struve, Otto 461. 

430. Mitchell, Miss 462. 

431. Somerville, Mrs. 463. 

432. Sheepshanks, 464. 

Miss 465. 

433. Ward, Mrs. 466. 

434. De la Rue 467. 

2 c 2 


Bond, G. P. 


De Morgan 
Beer and Mad- 

Terra Photogra- 




Straight Range 


Gwilt, G. 

Gwilt, J. 






Miller , 



Bond, W. C. 





468. Blanchinus 

469. Delaunay 

470. Faye 

471. Donati 

472. Alexander 
4V3. Janssen 

474. Cassini, J. J. 

475. Foucault 

476. Lee 

477. Piton 

478. Herschel, Miss 

479. Brayley 

480. Lockyer 

481. Daniell 

482. Grove 

483. Murchison 

484. Lassell 

485. Hermann 

486. Manners 

487. Schmidt 

488. Secchi 

489. Schiaparelli 

490. Harbinger Mts. 


Alphabetical Table of Lunar Nomenclature. 

Abenezra, 310 
Abulfeda, 305 
Adams, 437 
Agatharchides, 233 
Agrippa, 102 
Airy, 291 
Albategnius, 289 
Alexander, 472 
Alfraganus, 300 
Alhazen, 2 
Aliacensis, 296 
Almanon, 306 
Alpetragius, 205 
Alphonsus, 207 
Alps, 80 
Altai Mts., 315 
Anaxagoras, 168 
Anaximander, 174 
Anaximenes, 173 
Ansgarius, 349 
Apennines, 85 
Apianus, 294 
Apollonius, 8 
Arago, 63 

Aratus, 86 
Archimedes, 120 
Archytas, 46 
[Argffius, Mt.] 58 
Argelander, 467 
Ariadseus, 100 
Aristarchus, 148 
Aristillus, 83 
Aristoteles, 78 
Arnold, 39 
Arzachel, 204 
Atlas, 28 
Autolycus, 84 
Azophi, 309 
Azout, 6 

Babbage, 41 
Bacon, 360 
Bailly, 245 
Baily, 49 
BaU, 464 
Barocius, 359 
Barrow, 45 
Bayer, 235 

Beaumont, 322 

Beer and Madler, 446, 

Behaim, 350 
Bellot, 421 
Bernouilli, 21 
Berosus, 18 
Berzelius, 33 
Bessarion, 145 
Bessel, 73 
Bettinus, 251 
Bianchini, 138 
Biela, 392 
Billy, 266 
Biot, 329 
Birmingham, 463 
Blancanus, 260 
Blanchinus, 468 
Bode, 107 
Boguslawsky, 398 
Bohnenberger, 332 
Bond, G. P., 439 
Bond, W. C, 465 
Bonpland, 218 



Borda, 337 
Boscovich, 98 
Boaguer, 142 
Boussingault, 399 
Bouvard, 283 
Bradley, Mt., 89 
Brayley, 479 
Briggs, 163 
Buch, 361 
Bullialdus, 213 
Barckhardt, 19 
Burg, 50 
Biischmg, 362 
Byrgius, 279 

Cabeiis, 257 
Calippus, 76 
Campanus, 226 
Capella, 324 
Capuanus, 238 
Cardanus, 157 
Carlini, 128 
Carpathiis, Mt., 115 
Casatus, 254 
Cassini, 81 
Cassini, J, J., 474 
Catharina, 321 
Caucasus, Mt., 75 
Cavalerius, 155 
Cavendish, 286 
Cayley, 443 
Censorinus, 325 
Cepheus, 31 
Chacornac, 454 
Chains, 435 
Chevallier, 407 
Christian Mayer, 40 
Cichus, 189 

Clairaut, 357 
Clavius, 193 
Cleomedes, 12 
Cleostratus, 179 
Colombo, 333 
Condamine, 137 
Condorcet, 5 
Conon, 88 
Cook, 335 
Copernicus, 112 
Cordilleras, 274 
Coxwell, Mts., 405 
Crozier, 419 
Criiger, 278 
Curtius, 404 
Cuvier, 356 
Cyrilliis, 320 
Cysatus, 264 

D'Alerabert Mts., 27 
Damoiseau, 271 
Daniell, 481 
Davy, 209 
Dawes, 442 
Delambre, 301 
De la Rue, 434 
Delaunay, 469 
Delisle, 127 
Deluc, 194 
Democritus, 38 
De Morgan, 445 
Descartes, 304 
Dionysius, 99 
Diophantus, 126 
DoUond, 303 
Donati, 471 
Doppelmayer, 230 

Dorfel Mts., 246 
Drebbel, 240 

Egede, 79 
Eichstadt, 280 
Eimmart, 3 
Encke, 143 
Endymion, 27 
Epigeiies, 169 
Eratosthenes, llO' 
Euclides, 221 
Euctemon, 42 
Eudoxus, 77 
Euler, 125 

Fabricius, 383 
Faraday, 459 
Faye, 470 
Feniiat, 314 
Fernelius, 366 
5 Firmicus, 7 
Flamsteed, 223 
Fontana, 269 
Fontenelle, 171 
Foucault, 475 
Fourier, 285 
Fracastorius, 372 
Era Mauro, 219 
Franklin, 32 
Franklin, J., 418 
Frauenhofer, 317 
Furnerius, 345 

Galileo, 156 
Ganibart, 113 
Gartner, 37 
Gassendi, 232 
Gauricus, 185 



Gauss, 22 
Gay-Lussac, 116 
Geber, 308 
Geminus, 20 
Gemma Frisius, 363 
Gerard, 166 
Gioja, 44 

Glaisher, Mt., 406 
Goclenius, 328 
Godin, 103 
Goldschmidt, 427 
Grimaldi, 272 
Grove, 482 
Gruemberger, 265 
Guerik^, 215 
Guttemberg, 330 
Gwilt, G., 455 
Gwilt, J., 456 

Hadley, Mt , 87 
Hsemus, Mt., 68 
Hagecias, 391 

Hahn, 17 
Hainzel, 237 
Halley, 458 
Hanno, 382 
Hansen, 11 
Hansteen, 267 
Harbinger Mts., 490 
Harding, 152 
Harpalus, 133 
Hase, 342 
Hansen, 247 
Hecatseus, 351 
Heinsius, 190 
Helicon, 12^ 
Hell, 184 : 
Heraclides, 135 
Hercules, 29 

Hercynian Mts., 161 
Hermann, 485 
Herodotus, 149 
Herschel, 212 
Herschel, Miss, 478 
Hesiodus, 187 
Hevel, 154 
Hind, 457 
Hippalus, 225 
Hipparchus, 288 
Hommel, 387 
Hooke, 34 
Horrebow, 175 
Horrox, 460 
Hortensius, 119 
Huggins, 461 
Huygens, Mt., 90 
Hyginus, 93 
Hypatia, 317 

Inghirami, 244 
Isidorus, 323 

Jackson-Gwilt, Mrs., 

Jacobi, 395 
Jansen, 66 
Janssen, 473 
Julius Osesar, 96 

Kant, 302 
Kastner, 347 
Kepler, 144 
Kies, 214 
Kirch, 130 
Kircher, 252 
Klaproth, 255 
Krafft, 158 

La Caille, 292 
Lagrange, 281 
La Hire, 123 
Lalande, 210 
Lambert, 122 
Landsberg, 222 
Langrenus, 338 
Lapeyrouse, 348 
Laplace, 134 
Lassell, 484 
Lavoisier, 165 
Lee, 476 
Legendre, 353 
Lehmann, 241 
Leibnitz Mts., 259 
Le Monnier, 53 
Letronne, 224 
Le Verrier, 425 
Lexell, 199 
Licetus, 355 
Lichtenberg, 151 
Lilius, 394 
Lindenau, 370 
Linn^, 74 
Littrow, 55 
Lockyer, 480 
Lohrmann, 153 
Longomontanus, 192 
Louville, 141 
Lubiniezky, 216 

Maclear, 441 
Macrobius, 59 
Madler, 466 
Magelhaens, 334 
Maginus, 195 

Main, 436 



Mairan, 140 
Malapert, 258 
Manilius, 95 
Manners, 486 
Manzinus, 401 
Maraldi, 56 
Marco Polo, 91 
Mare Smythii, 424 
Marinus, 379 
Marius, 147 
Maskelyne, 67 
Mason, 48 
Maupertuis, 136 
Maurolycus, 358 
Maury, 440 
McClure, 420 
Menelaus, 70 
Mercator, 227 
Mercurius, 6 
Mersenius, 231 
Messala, 23 
Messier, 327 
Metius, 384 
Meton, 41 
Milichius, 118 
Miller, 462 
Mitchell, Miss, 430 
Moigno, 408 
Moretus, 262 
Mosting, 211 
Murchison, 483 
Mutus, 400 

Nasireddin, 198 
Neander, 373 
Nearchus, 390 
Neper, 9 
Newton, 256 

Nicolai, 393 
Nonius, 365 

(Enopides, 177 
Oersted, 30 
Oken, 380 
Olbers, 159 
Oriani, 14 
Orontius, 197 

Palitzsch, 341 
Pallas, 108 
Parrot, 290 
Parry, 217 
Pentland, 402 
Percy Mts., 416 
Petavius, 340 
Peters, 409 
Phillips, 423 
Philolaus, 172 
Phocylides, 242 
Piazzi, 282 
Picard, 4 
Piccolomini, 371 
Pico, 131 
Pictet, 181 
Pitatus, 186 
Pitiscus, 386 
Piton, 477 
Plana, 47 
Plato, 132 
Playfair, 293 
Plinius, 61 
Plutarchus, 15 
Poisson, 364 
Pollock, 449 
Polybius, 316 
Pons, 313 

Pontanus. 31 1 
Pont^coulant, 381 
Posidonius, 54 
Proclus, 60 
Prom. Acherusia, 69 
Prom, ^narium, 206 
Prom. Agarum, 1 
Prom.Archidseum, 452 
Prom. Lavinium, 450 
Prom. Olivium, 451 
Ptolemoeus, 208 
Purbach, 202 
Pyrenees, 331 
Pythagoras, 176 
Pytheas, 124 

Rabbi Levi, 368 
Ramsden, 228 
Reaumur, 287 
Regiomontanus, 201 
Reichenbach, 375 
Reiner, 146 
Reinhold, 114 
Repsold, 167 
Rheeticus, 104 
Rheita, 376 
Riccioli, 273 
Riccius, 367 
Riphsean Mts., 220 
Ritter, 64 
Robinson, 413 
Rocca, 277 
Romer, 52 
Rook Mts., 276 
Rosenberger, 389 
Ross, 62 
Rosse, 417 



Eost, 236 
Eumker, 428 

Sabine, 65 
Sacrobosco, 312 
Santbech, 336 
Sasserides, 183 
Saussure, 196 
Scheiner, 261 
Schiaparelli, 489 
Schickard, 239 
Schiller, 234 
Schmidt, 487 
Schomberger, 397 
Schroter, 106 
Schubert, 10 
Schumacher, 24 
Scoresby, 43 
Secchi, 488 
Segner, 248 
Seleucus, 162 
Seneca, 16 
Sharp, 139 
Sheepshanks, Miss, 

Short, 263 
Shuckburgh, 426 
Silberschlag, 101 
Simpelius, 403 
Sirsalis, 270 
Smyth, Piazzi, 411 
Snellius, 343 
Somerville, Mrs., 431 
Sommering, 105 

Sosigenes, 97 
South, 414 
Stadius, 111 
Steinheil, 385 
Stevinus, 344 
Stiborius, 374 
Stofler, 354 
Strabo, 35 
Straight Range, 453 
Street, 182 
Struve, 25 
Struve, Otto, 429 
Sulpicius Gallus, 

Tacitus, 307 
Taquet, 72 
Taruntius, 326 
Taurus, Mt., 51 
Taylor, 299 
Teneriffe Mts., 410 
Terra Photographica, 

Tbales, 36 
Theaetetus, 82 
Thebit, 203 
Theon, sen., 297 
Theon, jun., 298 
Theophilus, 319 
Timseus, 170 
Timocharis, 121 
Tobias Mayer, 117 
Torricelli, 318 
Tralles, 13 

Triesnecker, 94 
Tycho, 180 

Ukert, 109 
Ulugh Beigh, 164 

Vasco de Gama, 160 
Vega, 378 
Vendelinus, 339 
Vieta, 284 
Vitello, 229 
Vitruvius, 57 
Vlacq, 388 

Walter, 200 
Ward, Mrs., 433 
Wargentin, 243 
Weigel, 249 
Werner, 295 
Whewell, 444 
Wilhelm Humboldt, 

Wilhelm I., 191 
Wilson, 253 
Wolf, Mt., 92 
Wollaston, 150 
Wrottesley, 422 
Wurzelbauer, 188 

Xenophanes, 178 

Zach, 396 
Zagut, 369 
Zuchius, 250 
Zupus, 268 




Elements of the Moon. Epoch, 1st January, 1801. 
(Earth's equatorial diameter is taken as 7925*8 miles.) 

Mean longitude of moon at epoch 

Ditto node 

Ditto perigee... 

Mean distance from the earth (earth's radius 

Same in miles 

Maximum distance in miles 

Minimum do. do. 

Eccentricity of orbit 

Mean equatorial horizontal lunar parallax 

Maximum do. do. 

Minimum do. do. 

Moon's mean apparent diameter 

Moon's maximun do. 

Moon's minimum do. 

Moon's diameter in miles... 

Moon's surface in square miles 

Moon's diameter (earth's equatorial diameter 

Earth's equatorial diameter (moon's as 1) 

Moon's surface (earth's as 1) 

Earth's surface (moon's as 1) 

Moon's volume (earth's as 1) 

Earth's volume (moon's as 1) 

Moon's mass (earth's as 1) 

Earth's mass (moon's as 1) 

Density (earth's as 1) 

Density (water's as 1, and earth's assumed = 

Gravity, or weight of one terrestrial pound 

Bodies fall in one second in feet 

Mean inclination of orbit 

Maximum do. do. 

Minimum do. do. 

Inclination of axis 


118 17' 8"'3 

13 53' 17"-7 

266 10' 7"-5 






57' 2"-7 

1 1' 28"-8 

53' 51"-5 

31' 5"-l 

33' 30"-l 

29' 20"-9 















5 8' 


5 3' 




Rynodical revolution in days 29 '53059 

Sidereal do. do. 27-32166 

Tropical do. do. 27-32156 

Anomalistic do.' do. 27-55460 

Nodical do. do. , ... 27-21222 

Maximum evection l20'29"-9 

Maximum variation ... ... ... ... 35'42"'0 

Maximum annual equation ... ... ... 11' 12"-0 

Maximum libration in latitude 6 44' 

Ditto do. in longitude 7 45' 

Maximum total libration (from earth's centre) ... 10 16 

Maximum diurnal libration 1 1' 28"-8 

Surface of moon never seen (whole as 10,000, 

and diurnal libration neglected) ... ... 4198 

Surface seen at one time or other do. do. ... 5802 

Ditto do. never seen if diurnal libra- 
tion be taken into account ... 4111 

Ditto do. seen at one time or other do. 5889 

Mean revolution of nodes (retrograde) in days ... 6793-391 

Ditto do. do. in years ... 18-5997 

Mean regression of nodes per annum ... ... 19 21' 18"-3 

Mean regression of node between successive con- 
junctions of sun and rising node ... ... 18 22' 3"- 2 

Mean interval between such conjunctions in days 346-607 

Mean revolution of perigee (advancing) in days ... 3232-575 

Ditto do. in years 8-8505 

Mean advance of perigee per annum 40 40' 31"-l 

Ditto do. between successive con- 
junctions of sun and perigee 45 51' 23"-7 

Mean interval between such conjunctions in days 411-767 





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*. wtar , 



UN 9 1981. 

By the same Author. 


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